You may be able to come back on the at-fault driver in the accident (policyholder) for the expenses that go beyond the limits of his or her liability insurance.
It will depend upon many things such as state laws, statute of limitations and if you sign a release with their insurance company that does not allow you to pursue future monies or settlements from the insurance company or the policyholder.
Typically when you settle with an insurance company there is a release that you will sign. This document normally will release the insurer as well as the policyholder from future responsibility or liability in regards to the damages they previously caused. If you sign this release then you usually are not permitted to then sue the at-fault party since you accepted the settlement their insurance company offered.
Insurance and states laws do differ though so to find out if your state would allow you to take the at-fault party to court for monies beyond what their insurance company paid for the damages you sustained in the accident, contact a lawyer familiar with this type of case law.
Another avenue is if you have underinsured motorist bodily injury and/or property damage. If the other driver involved in an accident is underinsured, this coverage typically pays any difference between what the other driver’s insurance covers and what your bodily injury coverage will pay.