There are a couple of ways of doing this.
If you want an early appointment, book for the night before, call ahead and have them check you in. You should have until 1pm the next day with this method.
If you're booking the room that day or the day before, just tell them you'd like to check in around X time, will that work for them? If they get squirrely about whether or not that will actually be ok, just say "Ok, is it at capacity or not?", and they should have a clear yes or no.
There are almost always rooms available. If the above doesn't work, $20 to the front desk guy will