I don't know all the details, but after his self-titled sitcom ended its run on CBS in 1965, Jack Benny switched networks to NBC, for whom he made several specials, and appeared on Dean Martin's Celebrity Roast before his passing.
From his 1966 Christmas special comes an in-show advertisement for American Motors, the original makers of the Jeep brand. Jack is joined by Peter Hansen (General Hospital) and frequent pain in the tuchis (to Jack, anyway) Frank Nelson.
Now, to answer the question at the title of this post, we know that Benny did drive other cars, the Maxwell being part of his radio act. One of these times I'll have a bit that Jack did with Dennis Day for Texaco.