I tend to agree with the previous poster. We fly JetBlue, we leave the first flight and come home late. We park in the JetBlue lot (right by terminal 5) it's expensive, but it's safe, the car is safe and it's convenient. Let me know what you find.
Side note: We went November last year, and for some reason the car alarm tripped and the battery died. It was 11 at night, freezing (w/ 2 kids) and no idea what to do. The security people were there within 5 mins and jumped our car for free. So, another reason this is a good choice.