Thanks, all. The problem I had is I called the hotels directly (both in the Hilton chain and ones I stay at regularly) and asked about booking a room during the day. The operators I talked to said I had to talk to the front desk staff. I did that - they said I needed to call reservations. I did that and they said I had to call the hotels directly.
So I gave up and used a third-party site. Not a cheap option but better than spending 13 hours at the airport. I still haven't confirmed the booking is in the Hilton system though but will do so soon.