Doesn't Enterprise "pick you up"?
EWR long-term parking lots are inexpensive, and you can easily get to one of the many in-airport rental car locations via monorail/shuttlebus. Here's a map.
I've never used them, but there's a Budget Rent A Car office at 181 1st Street between Marin Blvd. and Manila Ave.
My biggest gripe with all of the rental car places in JC is that they are closed on sunday. You can return it on sunday but you will be charged for returning it on monday.
I've used Enterprise at A&P a few times over the years. My only complaint is that, like all companies, they never have the car your reserved. Once, for a 700-mile round trip, they gave me a huge Dodge Ram pickup. They said they'd give it to me at the cost of a Neon. When I told them I'd have to buy almost $100 more gas to make the trip, they reduced the price even more.