I've used mostly Reliable, but also InterCity, Horseless Carriage and a few others I can't remember.
I've had good experiences with all of them. I'm guess I'm probably lucky, with all the cars I've shipped/received over the decades. Never had any damage, or had to file a claim.
A company I most recently used (after all the big companies were booked up with the BJ auction) is Image Auto Group (702/278-9304). Mark Phillips is the owner. They are a small organization, but have nice 18-wheeler rigs (at least the one that delivered my car was) and are based in Las Vegas. I asked specifically, and they do NOT ever broker out loads to other carriers. They also put my car in the nose of the trailer, and it stayed there all they way until it got to my house....about 1800 miles. I realize that's not always possible, but I requested it, and they fulfilled my need.