If you were injured in a motor vehicle accident involving a rental car whose driver was not listed on the contract, you generally cannot hold the car rental agency liable for the damages. Though there may be limited exceptions to this rule, generally you will have to go after the automobile insurance of the driver, who should have his or her own auto insurance policy if driving anyway. If the driver is uninsured, you will need to use your own uninsured or underinsured motorist coverage on your insurance policy, if you have such coverage, to attempt to recover for your injuries.
Car Rental Liability
As a general matter, most car rental companies are meticulous about taking the information of any drivers or potential drivers. In addition, there are always stipulations in the rental car contract indicating who may drive a rental car. As such, the person who rented the vehicle and who violated the contract by letting an unauthorized driver operate the car, may be in some trouble as a result of your situation. Of course, the fact that this driver breached his contract isn't going to make the car rental company responsible for what happened to you.
If you could prove the rental agency was somehow negligent in renting the car (e.g. - perhaps they knew that the renter was going to let an unlicensed driver drive the vehicle), then you might have a case against the company. This, however, is relatively rare.
Getting Help for Car Rental Liability Issues
If you find yourself dealing with an accident injury caused by a rental car driver, you will most likely want to contact a skilled automobile insurance claim and personal injury lawyer to assist you with recovering the damages to yourself and or your vehicle. State laws regarding insurance can get quite complicated, and if you are serious about recovering damages, you will need to hire the best possible legal help to protect yourself.