If you are looking to connect to a domestic airport from JFK, your best bet will, by far, be LGA. HPN is located outside the city limits in Westchester County. Depending on traffic, this would be at least a 40 minute drive from JFK. Provided you could even get a taxi to take you there, it would be expensive. A car service would be expensive too, definately more expensive than going to JFK. At Noon on a Friday in July, you could hit traffic, but LGA is pretty close to JFK (within 10 miles). If you budget an hour for travel time between the two, you will, in all likelihood, arrive at LGA early.

Have you verified that there is no way to get from JFK to your final destination? JFK does offer a sizeable number of domestic flights. Even if it required a connection, it might be easier than transferring to LGA. With the new AirTrain at JFK, getting between terminals is now pretty easy.