I'm in Turkey right now and have had no issue having my hotels arranging airport transport, including today when I checked in at 2:00 PM and arranged a ride for 12 hours later.
If you do want to be closer to the airport for your last night, the Park Inn by Radisson Istanbul Airport, Odayeri (where I am tonight) is a decent place - but not a Hilton property.
It's in the middle of nowhere so I'd spend the day in Istanbul before heading out this way.