Well there are a few ways to go here:
1. Stay with the basic limited liability protections that is included with every move. Keep in mind this is not an isurance. It will compensate you $.60 per pound per article. So, if you have a table that weighs 100lb you will only receive $60 towards the repair of that item.
2. Some of the movers will offer insurance. Typically they would have a few options for you with different deductable.
3. You can choose a moving company and then just go with a 3rd party isurance just to name couple....moving insurance . com and baker international.
4. Your home owner insurance or renters insurance might cover your move so check with them before spending the money.