Not sure I'd want to be relying on a non-traditional car rental company at midnight.I think you really should turn to non-traditional car rental companies. In small towns, contacting a rental company at any time of the day is impossible, so you should look for alternative options.
I'd call the local (to the airport) for the rental agencies. Ask if they stay open to accommodate late flights. If not stay close to the airport, uber if necessary or take a car service to your destination and rent a car the next day.