'Consilience' is a term that goes back to William Whewell's philosophy of sciences; he argued that one of the signs of progress in science was the consilience of inductions:
It is true, the explanation of one set of facts may be of the same nature as the explanation of the other class: but then, that the cause explains both classes, gives it a very different claim upon our attention and assent from that which it would have if it explained one class only. The very circumstance that the two explanations coincide, is a most weighty presumption in their favour. It is the testimony of two witnesses in behalf of the hypothesis; and in proportion as these two witnesses are separate and independent, the conviction produced by their agreement is more and more complete. When the explanation of two kinds of phenomena, distinct, and not apparently connected, leads us to the same cause, such a coincidence does give a reality to the cause, which it has not while it merely accounts for those appearances which suggested the supposition.
Whewell took this to be the proper explanation of Newton's First Rule of Philosophizing, "We are not to admit other causes of natural things than such as both are true, and sufiice for explaining their phenomena," and, in particular, the 'true cause' part of it:
Newton’s Rule then, to avoid mistakes, might be thus expressed: That “we may, provisorily, assume such hypothetical cause as will account for any given class of natural phenomena; but that when two different classes of facts lead us to the same hypothesis, we may hold it to be a true cause.”
In a striking image he uses elsewhere, Whewell says that in consilience two different realms of inquiry 'jump together'. (As a side note, I think a great many works discussing consilience get Whewell wrong on a key point here. Whewell holds that Newton's First Rule, understood in consilient terms, provides a conclusive test and "will rarely or never" mislead us. People often point out that the very theory he uses as his example, Newton's theory of gravity, has since been superseded, and thus that his claims for consilience were excessive. But this idea, besides absurdly treating Whewell like a child incapable of recognizing an obvious possibility, assumes an account of what it means for a theory to be superseded that is completely inconsistent with Whewell's own. Whewell doesn't think consilience establishes a theory to be definitive in the sense of not being able to be improved upon. That wouldn't even make sense in Whewell's account of scientific discovery. What he means is that it is right to the extent that one is considering the classes of phenomena in question. In Whewellian terms, what happened when Einstein replaced Newton was not that the latter theory was proven wrong but that the latter's rightness was proven to be limited to certain conditions. There are no ruptures in Whewell's account of scientific progress; everything established properly remains true forever even though it may be so only in a limited way and may itself need to be explained. The problem arises in part because the people discussing Whewell have a flat conception of scientific realism that Whewell does not share: on Whewell's account, things can be true to a limited extent or to a degree of approximation without being false, because that's the regime we are always actually forced to accept in science, where our measurements, equations, and the like are never to perfect precision or without condition. Even if Whewell were wrong about this, his claims about a theory or hypothesis being made definitive or conclusive have to be understood in light of his own views on this point.)
The root idea of supererogation, of course, is paying more than is due; in moral terms, one does something good that is more than duty requires. Even characterizing this is often quite difficult in a number of modern moral theories, to such an extent that one can find people denying that there are any such supererogatory acts -- if you do something 'beyond what duty requires' you are either doing something immoral or something morally indifferent. This can all be summed up in the paradox of supererogation:
(1) Supererogatory acts are morally better than acts merely fulfilling duty, which are otherwise the morally best acts, and so are the acts that are morally best.
(2) Supererogatory acts are optional.
(3) Acts fulfilling duties are not optional.
(4) To do what is morally best cannot be optional.
Or, in other words, they are morally even better than doing one's duty, but not doing them is not morally worse than not doing one's duty, or even morally bad at all. The force of (4) is somewhat obscured in this summary; the basic idea is that our best moral reasons will favor doing what is morally best, so there will be at least an all-things-considered requirement to do whatever happens to be morally best. It's still wrong, but my purpose here isn't to talk about the paradox of supererogation.
One can see something of the analogy between supererogation and consilience by the fact that you could characterize consilience as being a case where a theory or hypothesis does supererogatory explanatory work. But I think we can say more about this by looking at how we might think of supererogation as a sort of practical consilience.
Obligations or duties are usually described by a deontic logic. I have suggested before that deontic logics should really be seen as logics characterizing solutions to problems -- that is, they are suited for characterizing what is required and allowed for a solution to a problem -- and that it is this very fact that makes them suitable for talking about obligation and permissibility. A moral duty can be understood as something that must be taken into account in a solution to a problem of moral significance (the 'problem' here may just be the problem of what to decide given the circumstances).
What would supererogation be, then? I would suggest that the most obvious possibility is that it supererogation would occur when, in facing a problem of moral significance, one resolves the problem and also other problems. This is not a complete characterization. But let us say for the moment you are engaging in a quasi-supererogatory act if you not only perform your duty in resolving a moral problem but do so in a way that resolves other problems of moral significance.
What more is necessary to make it fully supererogatory? If someone rejected that any kind of quasi-supererogatory act could ever be genuinely supererogatory, on what grounds would they do so? I think all such rejections would involve the assumption that all problems of moral significance are equally significant. (This is related to what in the literature on supererogation is often called the good-ought tie-up.)
This makes sense, if you think of it: if you have to solve, in some sense, all relevant moral problems at once, then there's no room for supererogation. Typical moral positions that reject the possibility of supererogation, like Kant-style deontologies or many utilitarianisms, do so because they essentially hold that there is only one moral problem (e.g., what is in accord with the categorical imperative) or that the solution that must be chosen is that which provides the optimal solution across all moral problems, understood in some unified way as one problem (e.g., achieving the best consequences).
Further, if you look at atypical variations of these that do allow supererogation, you find they also hold that moral problems are not reducible to one moral problem, and that they do not allow of any univocal measure that puts them all on the same level. For instance, John Stuart Mill's version of utilitarianism is an atypical kind of utilitarianism that not only allows for supererogation but also has supererogation as one of the central elements of the theory. This is possible because in Mill's Art of Life no one is required to maximize good consequences. People often fail to grasp this point: Mill explicitly rejects the idea that the principle of utility obligates in and of itself. Rather, the principle of utility structures all of practical reason (which is concerned with the good) and does so in three departments: Aesthetics, Policy, and Duty. An action becomes a duty only through being confirmed by multiple applications of the principle of utility. Roughly, something can become a duty when it not only contributes to the greatest happiness of the greatest number but does so in such a way that it contributes to the greatest happiness of the greatest number for us to punish those who don't do it. This is only one kind of moral problem, though.
Mill has a very good example, in his essay on Bentham, highlighting the fact that there are different kinds of moral problems in Mill's account. Brutus, the great Roman patriot, was once acting as magistrate, during which a couple of criminals came before him for treason. There was no question, nor room for doubt, that they were guilty of trying to overthrow the Republic through violence. There was also no question what the law required for their crime: death. And no one doubted what Brutus's duty, considered simply as a magistrate, was. So he did his duty and condemned them to death. The two criminals were his sons. And Mill does not deny that Brutus did his duty; but he notes that we can still reasonably say that his act was not admirable or lovable. There was something inhuman about it. Living a sympathetic, beautiful life is a different set of problems from living a life according to duty. One could, if one wanted, hold that only the problem of how to live according to duty was a strictly moral problem -- but it doesn't change the fact that the other problems are genuine practical problems, important ones, and even can be considered moral problems in a broader sense. Thus Mill's moral theory has plenty of room for supererogation: what would be ideal would be decisions that are moral (in the strict sense), useful, and beautiful. Forced to a choice, duty or morality in the strictest sense wins: these are higher priority problems. And you have no general duty to do beautiful and useful things. But part of what we take to be a good life is not just a life of duty but also a light of beauty and effectiveness, and thus actions that solve not just the problem of how to do our duty but also the problems of how to live beautifully and effectively are better than actions that only solve the problem of duty.
The way in which supererogation, framed in these terms, is like consilience should be more clear now by restating what a supererogatory action is in these terms: an action is supererogatory if in solving problems of moral significance that have priority, it also solves problems of moral significance that do not have this priority. (Priority is not necessarily a matter of importance but simply the kind of necessity that makes for obligation.) Thus supererogatory actions solve moral problems in such a way as to solve different kinds of moral problems, making them "jump together" in one solution.
There are many more things that could be said, but this post is getting a bit long.