Oh yeah. Been there, done that.
I'd first check out some classic car transport companies...several good ones come up on the first Google search response page if you search for "classic car transport services". Even FedEx shows up! Imagine sending your car home by FedEx.
I'd do that because while they might charge you $1000 (or whatever), the car is shipped door to door, insured, and typically shipped in an enclosed trailer. It's well protected and moved by people who do this for a living. I've had one Corvette shipped to me this way, and it was a great experience. As always, do some homework, check references, and all of that kind of thing.
The other aspect is that if you move it yourself, you have to fly cross-country (always an adventure these days), or drive, rent a trailer, and drive it back. The time off of work, lost vacation, and rental expense might rival what it would cost to have it shipped.
However, I've moved two cars this way, and I can only say the experience is highly recommended.
Seriously. I moved once from LA to Ohio (Air Force relocation) and towed my 1970 behind my rented UHaul, and I was grinning the whole way. It was an adventure, seeing the country (even if not from behind the Corvette's wheel), and I was surprisingly proud to have it with me. I got a lot of hand-waves and honks along the way.
I also recently towed a 1980 home, so the process is fresh in my mind. I'd sold my 1970 years ago (got married, had kids...you know the old story) but told my wife I needed another car to restore. I found this 1980 for cheap and hauled it home so I could do it single-handedly. I have to admit I saw that black nose filling the rearview mirror and I got a little emotional. So it was a cheap thrill that way too. Nice to have a C3 back in the garage (but I'll have another 1970 before it's all said and done!).
The place to go for a trailer is UHaul. Two reasons: 1) they have full-bodied trailers versus the front-wheel tow-bars. You drive it up (carefully!), secure it, and haul away. 2) UHaul still rents auto trailers, and there is no requirement for you to rent a UHaul truck in addition to the trailer, in case you drive your SUV East and tow with that. Many rental companies won't rent you a trailer without renting a truck as well due to their insurance costs.
And, with UHaul you can make a reservation and they'll have a trailer there waiting for you. Always
make that reservation. It would be a (bad word here) to go that far only to find they rented the trailer a day ago and the nearest available one is New Jersey.
If you go the trailer route, I envy you the trip.
It's a beautiful country we have, politics and current economy notwithstanding. Enjoy the scenery.
Some recommedations, though... Secure the car well, and double-check both the car and the hitch each time you stop for gas or lodging. I never once had a problem, but I'd want to know if there was one before the car slid off the back. I also covered the 1970 with a custom-fit car cover (Eckler's has them, as does everyone else). I wasn't so worried about road damage as I was keeping prying eyes away when I stopped overnight. In LA the car was hugely popular (once I secured a flag to the antenna mast on the Fourth, and the public response was amazing), but I found covering it helped keep it safer for some strange reason. At least it was never stolen, when several of my buddies had their cars stolen. I still parked truck and trailer under a light each night I stopped. And no, the cover never faltered, ballooned, or came off, even over a 2000 mile drive.
Best of luck!