Question: Almost all auto insurers I speak with require that I give them the model and make of the car I will be driving. For example:

* Chevrolet Silverado
* Honda Civic
* Subaru Outback

Consider if I got in a car accident when I was...

* driving a friend's car
* If I was on vacation driving a rental car
* if I was driving family-member's car when an auto accident occurs

Any insurance claim I file would be denied, because I was not driving a vehicle covered by my policy.

How do I get auto liability insurance in the United States to cover ***ANY*** car I am driving?

Answer: 
There is a kind of insurance like this, for people who don't own a car but nonetheless want to be covered for any car they drive, for example when renting a car or borrowing a friend's car. This kind of insurance goes under a couple of different names. When I first got it over 10 years ago, it was called "no-name insurance". But when I did a web search for that term just now what came up instead was "non-owner car insurance".
Auto liability coverage follows the driver, no matter whose vehicle is being operated, so long as the operator has the owner's permission to use the vehicle.

You will see the opposite stated in many places. But that is an answer to a different question. For example, say you are driving a friend's car and get into an accident. State law may require your friend's policy to cover you. In that case, you have no liability because your friend's insurance covers the accident. So the question of whether or not your liability insurance would have covered you if you did have liability becomes a purely academic one.

If you have liability, you have coverage. Someone else's insurance will cover it in most cases, so you won't have liability anyway.