There would be at least five pointers to locating a car mover you can trust who will not just take your money. One who'll take responsibility and look after you.
An independent business will be more likely to take an interest in your car than staff in an agent's office. So ask if your mover is an agent working on commission for a bigger business, or is an independent business.
It's possible things will go wrong during your more, and fixing this with an office near, or in, your new suburb will be less stressful than if it is across town. When you pick up your car, you'll get the service of local people if your mover has staff in offices around your new city. So, find out if the car mover has a network of offices around your new location.
There is protection, and quite likely better service, if your mover is a member of the American Moving and Storage Association, or a similar organization. Find out.
The better movers will not give you a quote immediately you inquire. They will want time to find out about your car, note things down, ask questions and then write a formal estimate that takes account of the particulars of your car. In other words, they will have a pricing system.
Lastly, the better movers will be open to your suggestions, explain clearly what will happen (and what won't) and answer all your questions about insurance, the move, any storage, and guarantees. Is your mover a man like this?
It's obvious you should never hire your car mover solely on price. Cheapest is probably not the best. Look instead for a mover who will truly care for your vehicle.