Are you stopping overnight when you move? If so, then you should get an enclosed trailer that you can lock. Even that's not a guarantee of security but better than leaving everything exposed. A covered trailer is also better for rain storms. A tarp can only do so much.
We've traveled with a rental enclosed trailer of furniture. We also own a utility trailer that TCS made.
We rented the enclosed trailer after my mom died in CT. We needed the trailer for one way only, to bring back her furniture to our house in SC. However, it was cheaper for us to rent it empty in SC, drive up to CT, load it up, and then drive back to SC. That's because of the return charge for North-to-South one-way trips. It was weird that round-trip fee was cheaper than one-way fee.