Recently I moved one 56 from Palm Springs East to Nj and the other from Nogales Az to Nj I used a small trucker for Palm Springs who advertised in Hemmings..he got the car there ok and the price was decent but his equipment was basic and older.. that car made a run for it when the strap holding it snapped and it rolled down a hill almost going through my neighbors den.. then rolled almost out the back wall of my frame barn, the driver was unmoved ..and barely got enough feeling for what almost happened all he could say was" imagine that " I used a bigger company the next time, also out of Hemmings .. they called me on a Holiday to get over to my barn at 11 pm to take my car and pay them.. never having called me to tell me they got the car. i refused and they reluctantly had to move it the next morning.. I noticed how rough they were with other owners cars actually using someones SUV to drag the Imp up the hill .. they left it in a hurry wanted their cash and off they went.
So my answer would be make sure they have references and that they will call when they get it and tell you before its arriving and that they have professional newer equipment..Not all of us enjoy the panick of a runaway 56 Imperial Cpe..