At Airport locations it is easier to just bring your car back and grab the new one in the stall.
At Local Editions, they are usually more then happy to let you keep your car. I had a nice Sable YF with everything (Sirius, 6-disc changer, sunroof, leather, automatic climate control) I rented daily for two weeks, followed by a nice Impala for another week (the Sable needed an oil change so I had to swap). They just closed out the rental each day and handed me a new contract. Took all of three minutes. Worked great and I was 21 rentals closer to PC. Nice thing was, it only ran me about $10 more a week to do individual rentals then weekly (which only would have earned me three).