I've done piggyback rentals like this before when I have a personal/business combination and want separate receipts.
What I usually do is drive to the airport, but not the Avis return line, and walk in to the Avis counter. Give them the contract and ask them to close it out, and open your new rental with the same car. I haven't had a problem doing this... it saves them from cleaning the car and you from moving your luggage. However, I only do this when the car I get the first time is decent.