I’d be happy with a few thousand dollars, I’ve read nightmare stories where they suddenly increase the price to tens of thousands of dollars mid move and hold your stuff hostage.
I sold my pick up so I am looking at one of those companies that drop off a shipping container at your house for you to pack yourself. Then they take it to your new place.
Also, if possible, have friends and family handy to take photos and note any 'dings' noted on the paperwork by the movers. Every scratch, dent, stain is marked on those inventory sheets without any clarification on the extent of the damage. When the items are unpacked, if the damage exceeds the original state of the item, it's tough luck unless you can prove the damage was done by the moving company.
We are currently seeking compensation for my daughter's last move where the mover's destroyed her brand new crib. We disassembled it and handed all the components to the movers to pack appropriately. They delivered each rail with separate split spindles and the drawers cracked in half...insisting the damage was done prior to pick up. We have photos, but await final determination on any reimbursement.
GOOD LUCK with your move.