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: 
Ask insurers directly about "named non-owner" policies. They're uncommon, not all insurers offer them and those that do have to manually underwrite and manually quote the rate. It's not going to show up in any automatic tools.
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.