There are certainly places you could stay - you could drive say an hour to Norwalk, CT and stay there - my only consideration would also be if your plane is delayed, and now it is even later, by the time you rent a car and actually get on the road - is it worth it?

I think I'd rather get up and head out immediately after rush hour traffic, but that's me....

If you do want to get a jump on things, Norwalk might not be a bad place to head.

The hotels at JFK are not great but I can certainly understand not wanting to drive very far at that hour. If you do that, the Courtyard at JFK has been reported as being a decent place to sleep by NeoPatrick.

While I know you decided to forget the ferry, given the information on your arrival time, you might think about doing the drive to Port Jefferson when you arrive and taking the ferry on Thurs. morning. It cuts off a big chunk of the drive on I 95 and Port Jeff. is a lot nicer spot than JFK. The drive is about 1 hr. 15 mins. to Danford's Inn.

I also found a car service from Newport. You might get a quote from them if you don't feel like driving.
http://www.airportexpressdirect.net/services.html

I think escargot has a good idea. Norwalk is about an hours drive from JFK. You could stay at the Doubletree hotel on RT 1, just off 95 at exit 13.

This is a better quality hotel than what's available at the airport, and you'll miss the morning JFK/NYC traffic. In fact, heading east from Norwalk at that time you will probably be in the clear, and should be in Newport in well under three hours. Or you could stop off in Mystic, (about an hour and a half away), see the town and have lunch, then continue on.