The terms of an uninsured motorist property damage (UMPD) policy as well as state insurance laws governing this type of policy differ. Some insurance companies UMPD coverage allows for claims to be placed when an unknown person has hit your vehicle and left the scene of the accident, a hit and run. While others require you to show proof that the person that struck your vehicle was uninsured.
In some states such as Louisiana a hit-and-run driver may be classified as uninsured motorists if there is a disinterested witness to the accident. With the witness stating they saw a vehicle, whose owner and operator cannot be identified, striking your car it should enough for the insurance company to take the claim.
With UMPD coverage, there are some insurance companies that will not provide coverage for property damage from a legally liable uninsured if the person is an unidentifiable motorist (e.g., a hit-and-run driver) so that is why it is important to check the terms of your policy and speak to your insurance company to find out if you can place a claim for the damage your car sustained in the incident you described.