Well, Page 106 of Your Story says either at the end of the scene (out-of-conflict), or at the end of the exchange (in conflict), so you might be right.
Personally, mind you, I'd always judge it to be at the end of the scene. Conflict tires people out ... having someone invoke your sprained ankle on their attack should not make you stronger for the followup attack three seconds later just because it's a new round (exchange). As a compromise, you could say that the target gets the Fate point at the end of *their* next exchange, which could be the following round of combat.