Aquinas's Third Way is often treated as a form of what later came to be known as an argument from contingency. This is perhaps natural, because it makes use of concepts of necessity and possible-to-be-and-not-to-be, but is in fact not true. The reason for this is that the conception of necessity and its opposite are very different from what later is characterized in similar terms. In Quodlibetal Question X, question 3, article 2, Aquinas has an objection and reply that is useful for clarifying the real meaning.
The question is whether the rational soul is corruptible. The second objection argues that it is:
Something with the power to exist always does not exist at one time and not at another (for a thing exists as long as its power demands). Hence anything that exists at one time and not at another lacks the power to exist always. But everything that begins to exist, exists at one time and not at another. Hence, nothing that begins to exist has the power to exist always, and thus nothing that begins to exist can be incorruptible. Yet the rational soul begins to exist. Therefore, the rational soul cannot be incorruptible.
[Thomas Aquinas, Quodlibetal Questions, Nevitt & Davies, trs., Oxford University Press (New York: 2020), p. 138.]
Aquinas responds to this:
The argument is the Philosopher's proof in book I of On the Heavens that every generated thing is corruptible. But it applies to things that come to exist and cease to exist by nature: their lack of power explains why they cannot always have existed and will not always exist. The argument does not apply to things that come to exist by creation: God gives such things the power to exist always, although they cannot exist before being given that power.
[Ibid., p. 141]
That the Third Way is connected to Aristotle's argument in De caelo for the nature of the entire universe was shown by Lawrence Dewan quite some time ago, but it seems not generally to have diffused out. As Aquinas's reply indicates, a key concept of the De caelo is that generability and corruptibility are coextensive; what is generable is corruptible (and vice versa), and likewise what is ingenerable is incorruptible (and vice versa). The generable/corruptible is something that has the power to-be-and-not-to-be; the ingenerable/incorruptible is something that has a power to-be-always. The Third Way is an argument, based on this, that there must be something with power to-be-always; as Aquinas notes in the reply, the power to-be-always can be 'given' to something -- i.e., it's not an abstract possibility but one that is based on existing, so something has to exist before it can have it. Either it has always existed in itself, or it depends for this power to exist always on something that ultimately has to have always existed in itself.
Thus we can gloss the Third Way in the following way:
The third way is taken from the possible and the necessary, and is as follows. We find in things those which are possible to be and not to be, because they are found to be generated and corrupted, and consequently are possible to be and not to be.
The possible and the necessary here are in fact the generable/corruptible and the ingenerable/incorruptible.
It is impossible that all the things that are be such, because what is possible to be and not to be, sometimes is not.
An existing thing that is generable must at some time have been generated, because a thing cannot be generable if it has always existed.
Therefore if all things are possible not to be, at some point there was nothing in things.
This is the point in the argument that often trips people up. Why couldn't there be a series of things 'possible not to be' that caused each other in an overlapping way, infinitely back? But this worry depends crucially on the false assumption that a series of existing things is not something existing. If there existed a series of existing things that itself has always existed, the series would not be possible not to be -- it would be something that always existed, and therefore necessary in the sense used in the Third Way, which would mean that not all things are possible not to be. Thus if the series of things has not always existed, at some point there was nothing; if the series of things has always existed, there is something that did not begin to be.
But if this is true, then now there would be nothing, because what is not, does not begin to be save through something that is; if therefore nothing were a being, it was impossible that something began to be, and so nothing would be, which is obviously false.
Causes that do not in any way exist do not cause anything; so if there was nothing, nothing could be caused to exist.
Therefore not all beings are possibles, but something must be necessary in things.
That is to say, if there are generables/corruptibles, there must be something ingenerable/incorruptible.
But everything necessary either has a cause of its necessity or does not have one.
That is, what exists always must exist always either because of itself or because of something else. This is the point that Aquinas is making in the response in the quodlibetal question.
But it is not possible to proceed infinitely in necessaries that have a cause of their necessity, just as was proven for efficient causes.
The argument in the Five Ways that comes closest to being an argument from contingency is the Second Way, which does tie in because the Third Way refers to the Second Way; this is another element that makes the Third Way look like an argument from contingency. However, the element of the Second Way that is used here is specifically the structure of its argument against an infinite regress of causes.
Therefore it is necessary to posit something that is necessary through itself, not having any other cause of its necessity; but what is cause of the necessity of others, that all call God.
A naturally always-existing cause of other things being able to exist always, in other words.
One of the important aspects of the Third Way is that it more directly considers the entire universe -- unlike the First Way, whose primary consideration is changed things, or the Second Way, whose primary consideration is series of causes, the Third Way considers things that are such as to be always or not always, and as the roots of the Third Way in Aristotle's cosmological treatise, De caelo, indicate, this touches on general considerations of the nature of the universe as a whole. Either the universe is such as to be generable/corruptible, or it is not; if it is, it depends on something else that is ingenerable; if it is not, it is itself ingenerable, and its ingenerability is due either to its ingenerably existing by nature or being such that it ultimately has ingenerability from a cause that ingenerably exists by nature.