For what it is worth, something I've done at least twice with rental cars is buy a basic trunk rack and donate it at end of my one-way car rental. Often the cost of the rack is less than the difference needed to get a larger vehicle.
I had never considered that. You might even find something used at Play it again sports or something. I have a Saris "Bones" type rack that had $12.99 stickers on it that looked lke they came from a second hand store. It was in perfect shape when I picked it up from the curb for free.
Any way the Bones rack easily breaks down into small pieces that would pack small and easily mail ahead or home. I could see using this rack by mailing it to yourself by general delivery mail in a city where you needed to travel from or mailing it home or ahead from the start of a section your tour.