Can I get compensated for my out of pocket expenses that I paid that the person's insurance didn't pay?
We cannot give legal advice. The following is not a substitute for the advice of a local attorney. But we hope the information will be useful.
Generally, if your property is damaged in an auto accident, and it was caused by the negligence of another person, they are responsible for your damages - all expenses related to the incident, in order to put you back into the same position you were prior to the incident. The other person's insurance company typically pays you a check to cover these expenses (repairs, etc).
What you seem to be telling me is that after receiving such a check, you had additional expenses, and you want these reimbursed as well. You are going to have a much harder time getting paid back for these, because when you accepted the first insurance check, you likely signed a settlement agreement stating that you were accepting the check in full settlement of the incident, and agreeing not to seek any more money.
If this is not the case, then just send the bills to the representative who cut you the first check, and presumably they'll cut you another check.