Can't say about Chattanooga, but in Knoxville there is a Jameson Inn near the airport, with free shuttle (it is a taxi that they provide a voucher for)... very nice, clean place and they are happy for you to leave your car for free for almost any length trip if you are staying either before your flight out OR the night after your return. We have used them twice when flying out of Knoxville and they could not have been nicer. The cost of airport parking for the week would have been more than we paid for the room (and we got the room and shuttle to and from the airport too).