I haven't done any enclosed transports but I've shipped 3 cars, 2 cars with the same company and they do a good job, Countrywide auto transport. My friend is currently getting his car shipped with them as well but is having timing issues with them. I'd check em out...WHATEVER YOU DO...
DO NOT SHIP D.A.S. auto shippers. They gave me the biggest run around i've ever had with a shipping company, only to find out they wouldn't ship my car, so I had them cancel the order and I booked another company, though they still charged me for shipping my car though they never followed through!!!!!!!
Stay away from them.