I previously noted a supposition that allows subalternation without any bothering with existential import; discussing the matter with Tom, I said at one point that I didn't know if the particular supposition I noted was required. Having thought about the matter more, I can think of the following distinct suppositions that allow for subalternation.
(1) A and I as having existential import. Whatever it means to attribute existential import to propositions, it's generally taken for granted that attributing it to A and I allows subalternation. Similarly for E and O.
(2) Some S is S. Whether we take "Some S is S" as existential or not, it still makes supposition possible:
(3) A propositions as double propositions. Lewis Carroll allows subalternation in his system by making A propositions double propositions: "All S is P" simply means "Some S is P and No S is nonP." (He also accepts that A and I have existential import. But I don't see that this is strictly required by the move; Carroll accepts existential import for A and I for independent reasons.) Welton also has this view, and suggests the same for E propositions.
(4) Instantiation with generalization. We can add to term logic forms of instantiation and generalization:
Start with -S+P (All S is P, i.e., every instance of S is P)
instantiate to *S+P (a given instance of S is P)
generalize to +S+P (Some S is P).
Whether you think this fourth supposition is a case of 'existential import' depends, I think, on what you think something like universal instantiation is in the predicate calculus.
I'm sure there are others that could be put forward.
UPDATE: Doing a bit of reading, I find that Carveth Read claims that subalternation follows merely by the principle of identity; McCosh says it follows from the principle that "whatever is true of a class is true of any and of each of the members of the class." I think Read means same thing as McCosh; Veitch also claims subalternation rests on the principle of "Identity of whole and part". Morell likewise says that the correctness of subalternation "depends upon the relation which a logical whole bears to its parts." Parimal Kumar Ray gives two arguments for subalternation: (1) the particular simply repeats (part of) the information in the universal; and (2) what fails even in one case cannot be universally affirmed and what obtains even in one case cannot be universally denied. Francis Garden just says that it is "plain to the dullest capacity."