Bill,
Honestly the best way if to try Uship.com and weed through the brokers until you get an actual driver or owner/operator.
I worked on Joe Locandro's 60 convertible, prepping the car for a RT 66 tour from Chicago to The Pacific ocean. The car was picked up in my area near Phila, PA and delivered to Chicago. Joe's friend, a non car person, was there to receive the car at his residence in Chicago.
The lower right front fender has a 10 inch scrape at the exact spot where the little wheelwell hump inside the trailer is. It wasn't noticed during the unloading and the person signed off on the delivery which absolves the driver of any wrongdoing. The guy loaded the car very carefully and bent my ear for 30 min about how careful they are with collector cars and seemed like a great guy. But yet, the car is scraped in a spot that can only have been what I suspect to be the inner wheel well inside the trailer. 1960 Caddy's are VERY wide and don't even fit well in large car carriers.
So you take your chances every single time one of these cars is moved. I hate dealing with brokers because you never know who is getting the load, how many times it's loaded and unloaded, what it's parked under, etc.
I have shipped and received a lot of cars. I have had them come to me with fluids dripped on them from cars above, but the driver swears the car was covered up. I have gotten convertibles shipped with the top down and grease on the seats from a trucker climbing in or out. I sent a car to California that was in the top front position on an open car carrier and had damage to the roof.
I guess my answer isn't very helpful but at least you know what to look for.
Brian