By February 1960, Hertz was advertising cars in Flying, but no word on what happened to the R-A-P. There was an ad for an FBO (Dupont at MIA) offering Hertz Rent-A-Plane service. That was the last mention of it in Flying.
Another "lease-a-plane" operation stared in Illinois in 1970 but it didn't appear to have gone anywhere either.
There is one interesting federal lawsuit. Someone rented an twin via Hertz from the above FBO to fly arms to Cuba in 1958:
On August 16, 1958, C. W. Hormel rented from duPont, using a Hertz Standard Rental Agreement, a twin-engined 1955 Aero Commander airplane to be flown by Hormel's pilot, Guillermo Verdeguer, for the stated purpose of transporting Hormel from Miami, Florida, to Birmingham, Alabama. The plane, flown by Verdeguer, flew over Cuba with an overload of arms and ammunition aboard intended for Cuban revolutionaries. The plane, while over Cuba, was fired upon by the armed forces of the Cuban Government and it was unable to land. The plane was then flown several hundred miles, and, being out of fuel and having an engine disabled, perhaps from gunfire, it was ditched in or near Guantanamo Bay. The crash resulted in a total loss of the aircraft. The crash, so the jury found, was caused by the action of the Cuban Government in defending against insurrection and rebellion while the occupants of the airplane were engaged in flying arms and ammunition for the purpose of the revolution.
This ended up in federal court where the FBO sued their insurer for failure to cover this incident.