Axiom 4 in System S4 is this: ◻A -> ◻◻A
Its corollary is this: ◇◇A -> ◇A
As I understood with the corollary, it means that "If the fact that A is true in at least one world, true in at least one world, then A is at least true in one world." Which seems plausible.
But then I guess its converse (◇A -> ◇◇A) would also be true. Since it would mean that "If A is true in at least one world, then the fact that A is true in at least one world is true in at least one world too." Which also seems plausible.
Thus the question is this, is my assumption true? Is the converse of (◇◇A -> ◇A) true?
It's already implied by ( $◻◻A → ◻A$ ), which is a simple consequence of (M), namely the axiom schema ( $◻A → A$ ). (M) is included in S4. And yes your intuition about necessity and possibility is also correct.