Last year I used Enclosed Vehicle Transport,
www.enclosedvehicletransport.com, 888-827-6799 out of Buford GA.
I needed to transport a new car from Atlanta to Maryland and didn't want to go with a carrier who only had big multi-vehicle carriers. There is less opportunity for damage if the vehicle is loaded once and unloaded once. Multi vehicle trailers will cost less but I don't like the idea of loading and unloading a prized vehicle multiple times as pick up and drop offs are made. In addition, some of the carriers sub out the work so it was important that I knew exactly who the driver was, that they were the only ones hauling the car and it wasn't going to be unloaded several times as cars previously loaded are taken off the trailer for delivery.
In the search Enclosed Vehicle Transport was highly recommended and they have 1 or 2 vehicle trailers.
The entire transaction with Enclose was professional and mistake proof with communications being top notch. I was even able to talk to the driver, he called mid point to give me an update on the approximate time of arrival closer to the actual time.
If it's lowest possible price, you will give up some of this type of service for sure based upon my experience and research. FedEx and a few others will give you an approximate pick up date and drop off date but will not likely be able to guarantee those and can't say if the vehicle will be loaded and unloaded multiple times.
Good luck.