If I was at-fault for a car accident and the person I hit accepted payment from my insurance, can he get any more money from me?

There are two issues or questions here:

1) In accepting money from your insurer, did he sign a settlement or release agreement of some kind? (He should have; if not, your insurer may have erred in not protecting you, and it's possible you'd have a claim against them.) If he did sign such an agreement, and if it bars him from suing, that agreement is enforceable; in this case, he would have signed away his right to sue. You should ask your insurer if he signed any release or settlement, and, if so, get a copy.

2) If he didn't a settlement or  release, he can't double collect for the same loss, cost or injury, but could sue for different damages from the same accident. For example: say that he suffered $3,000 of car damage and $7,000 of medical. If your insurer paid him $3,000 for the property damage, he could sue you for the medical.

Or if his total damage, injuries, costs, losses, exceed your policy limits, he could sue for  the excess. Say you had $25,000 of liability coverage, but he suffered $50,000 of injuries. The insure is only obligated to pay up to the policy limit, but the injured person could try to recover for all his losses. So if he had $50k of injuries and the insurer paid out your whole policy limit of $25k, he could sue for the $25k they did not pay.