Why wait until their next turn then? They’ve declared an action. It’s the DM’s job to narrate the result of that action. To me, the obvious result of making such a leap is that you land on the other side. Of course, the character gets extra movement that way, but if you consider it permissible then why not resolve it?
My question is if it’s really a permissible action declaration, given the player doesn’t have the resources (feel of movement) needed to perform said action. Phrasing the declaration as if it only applies to the initiation of the action and not the obvious result just seems like an attempt to game the system to maximize movement.