First decide what you plan to pay, business class or tourist class.
Nearly all of the companies you will find advertising in the price-competitive market are brokers. They take a cut, then put the run up onto dispatch board for independents or companies to grab. Most of these run open transporters, some do have premium services on offer.
Right now things are not great in that industry.
Have a look at
http://www.transportreviews.com/ but don't take what you see there as gospel.
I have found that they are all sweetness and light when you first call. Once you pay and the job is dispatched, you may find the tone changes. You may wonder who is paying whom.
If the car is valuable, or you wish to be assured first class service, then look for one of the companies that deals directly with owners and transports rare/restored cars on covered transporters. However they will be at least 2x more then open.