Check out web site avistarparking.com.
Avistar is an off premises long term parking that has locations at all the NY area airports. Gram and I use them all the time as does almost everyone I work with. They are safe, secure, efficient and less money than airport long term parking. There are discount coupons available on the web site.
You can also try Corporate Parking. We drive to the lot, they take us to the gate in a new van, wash your car while you're away, and have your car ready in the driveway when you return. Smooth, inexpensive and safe operation.
We've used them for years and put out recommendations, as well. Their number is 908-354-8855. They do have a 1-800 number, but I don't have it on hand.