An umbrella policy protects your stache if you are found at fault for something. The other aspects of your auto policy are for if someone else is at fault, so not related. But, as
@seattlecyclone says, you should coordinate your auto and umbrella limits to make sure you don't leave a gap.
So, umbrella amount >= stache amount. (Because they only go in so many increments, and change it so often) Then if you are ever sued "for every penny you have!" then instead, the insurance company's lawyers show up. As was also said above, states can vary on laws about exclusions, like primary residence, retirement savings, etc. so make sure you are talking with someone who can speak to that.
There is some quirky behavior by the insurance companies that is really telling about what kinds of events trigger umbrellas. Very few companies will write a standalone umbrella policy; they typically coordinate with other policies, so that can be difficult across companies. State Farm wouldn't even write us one without a car, even though we own our home. I always though slip and fall type cases were a big deal. Of course, that might be rare in Houston. When I was working overseas, a $2M umbrella was part of my expat insurance bundle from Chubb, along with car and home.