Before you sign the contract, you should also reach grounds on the insurance offered by the auto transport company. Your car should be covered against any damage that may happen during the transport. You must also inquire if the insurance policy would serve as the primary or secondary insurance.
Proper verification should be done to find out if there are any restrictions or exclusions to the coverage while the car is being transported. Make sure that whatever assurances the company gives are collected in written format than oral promises.