I used DDP High Desert Equine Transport out of Oregon, when I had my mare transported from SC to WA a month ago.
They are a husband and wife team. They did awesome with my horse, and she with them. They kept me updated regularly on her well-being, where they were in the trip, etc.
The thing I really liked about them is that they offload the horses at night to individual paddocks or stalls at barns they are acquainted with across the country. So the horses really get a chance to stretch their legs!
The price I paid for them was right in line with all of the other companies, but they struck me as a bit more personable and horse-friendly. She is very knowledgeable regarding illnesses/travelling issues and such in the horses, also. The trailer has remote controlled AC too! Great for the south and the desert areas!
I believe you can just send them an email and they will get back to you regarding shipping your horse.
DDP High Desert Equine Transport LLC