It's hard to know exactly what the best way to write your paper is without more details. But I can see the following structure being commonly and effectively used:
Theorem If [condition P] then [conclusion Q].
Proof: Blah blah blah. $\square$
Remark: We see from the above proof of the theorem that it in fact suffices to assume only [weaker but more technical condition P'] to reach the same [conclusion Q].
In other words, rather than writing what you want to say as a "corollary to a proof" or a "porism" or a "scholium" or any other 10 dollar word, just put it there as a remark. I view a remark as any standalone comment which provides interesting context for the rest of the paper, but which is not seriously needed in anything that follows.
Incidentally, I disagree with those who say it is important to state the stronger but potentially more subtle or cumbersome to phrase conclusion before the proof. Stating a theorem in a clean way, giving the proof, and then explaining that it is visible from the argument that the conditions can be weakened somewhat is often a more natural flow of ideas.