I have used CEVA Logistics, (used to be TNT car carriers).
Salisbury to Newcastle for my Oldsmobile. Trip was 7 days, was actually shorter because the car was at a depot for a day or two.
CEVA also took a car I sold to Western Australia. Fast and efficient.

I found with the smaller companies I had to wait and wait and wait for a back load either way.

Mind you, these cars were driveable, I don't think they do cars that cannot start, stop or have to be winched on board.