In Texas, the law requires you to have liability insurance, but that won't pay to repair your car after an accident. For an atfault driver who is underinsured, you must first file a claim against their liability insurance.Texas law requires drivers to have liability coverage to pay for damage to other people's cars in the case of an accident. Report the accident to the police. Understanding Your Rights: First and foremost, you have the right to seek compensation for your injuries, even if the other driver is uninsured. If you have suffered injuries from an accident with a motorist who has no insurance or too little coverage, call our Texas attorneys for help today! If you're a victim of a hit-and-run accident, you can file a claim against your uninsured motorist coverage. Hi, I've parked the car in my apartment parking lot and someone hit and run in the night. Further, in some instances the other driver may not have adequate car insurance. How Does Uninsured or Underinsured Motorist Coverage Work in Texas?