You can and should get insurance coverage for your belongings for any move no matter how short or long the distance. It’s easy for things to break accidentally, even with the most careful movers.
For a fairly standard residential move, you can look into full-value protection, but if you are moving expensive and valuable items, you might want to look into third-party liability and protection. The best nationwide movers will automatically insure you with basic coverage at around $.60 per pound.