A large state, no, but if the OP is headed to the Hartford area or the southeastern part of the state, it can easily be a 2-3 hour drive from JFK. My car service charges $275 one way between Hartford County and JFK. About $2.75 per mile.
Another option for the OP to consider is renting a car at JFK and driving. It should cost you less than $100 and the rental car facility is relatively easy to access from the terminals via the Air Train.