"I’ve never personally tried to get a player interested with a scholarship offer alone." Good, because the offer of a scholarship alone doesn't raise the recruits interest at all. On the last day, the recruits that are automatically opened up for offers could sign with just the offer if no one else has put any recruiting effort into them.
To the OP, you must rescind and take the chance that you will subsequently fill the schollie. I would give two or three equivalent players the final offer, since if you just offer one you could easily be out-recruited on that guy. Tip the odds in your favor by preparing several players for that last schollie.