Hoare logic - partial/total correctnes and strength invariant
You're misunderstanding.  It's not that when you try to prove total correctness, Hoare logic somehow magically finds a stronger invariant for you.  Hoare logic doesn't find the invariant for you -- either way, you have to find the invariant yourself and then prove it.

Rather, it's that you need to prove a stronger invariant if you want to prove total correctness.  A weaker invariant can suffice to show partial correctness.  Therefore, total correctness can often be harder to prove than partial correctness (proving total correctness requires more work, because you have to prove a stronger, more detailed invariant).