This is something I deal with on a regular basis with my occupation... Most shippers are brokered out, so you don't really know who has your car, and some people are ok with that. However, depending on how valuable the car is and how concerned you are about it, many folks prefer to use a reputable independent carrier where the company has their own trucks and drivers, which is becoming more rare these days. You also have the one man operations and I know 4 good ones, but the problem is they are highly sought after and usually booked for 4 to 8 weeks out. I know of 4 independent larger carriers, but they all run east coast only so that won't help you. The only larger independent shipper that I recommend to my customers for coast to coast is Reliable Carriers, here is their website... https://reliablecarriers.com/
PS: You could also message blackrams and see if he is bored and willing to make a run like that, but I don't think he would haul coast to coast plus empty miles.
[This message has been edited by Skybax (edited 06-09-2021).]