You might find a free leftover cart sitting around long term parking before it gets collected. Not sure if there is a stall there to rent them. You can take the carts on AirTrain.
Following up in case anyone else has the same question: there are multiple smarte cart rental stalls in LT parking--follow the signs to find them. As I recall, there's one stall in each half of the garage (i.e., one on each side of the skyway). I also found several laying about on the 1st floor of the garage near the elevator, so check there first if you want to snag one for free.