Most removal companies give insurance for cases of damages on your belongings. However, this insurance is usually very minimal so you would need to get additional insurance just to save yourself from a lot of worry and stress. And although many of these removals company also offer supplementary insurance for additional fees, it is advisable to get it from independent insurance companies.
If you have the budget, getting full value protection or lump sum value would be a good idea.
While the first insures all kinds of damages and losses, the latter insures your things up to the value declared on the bill of lading. These two will afford you the most coverage but then again the costs are also higher. What you can do instead is to get more coverage for your belongings with the higher value and the others with transit insurance for instance.
Take note that removals companies do not insure goods that are not packed by them. In making claims, make sure you do not unpack the boxes that contain the damaged goods. Corrosives and flammable substances would also invalidate the insurance so make sure you do not pack such items with your things.
Before purchasing additional insurance, make sure that you understand what it covers and what the terms of the insurance are, this way you can be sure that you are getting the right insurance for your things.