Personal policy but using car for "work" on call

We have a car with full coverage through Nationwide/Independent agent. We do not use it for Uber/Lyft or any "business" use the problem is my wife has certain days of the month where she has to do an oncall (crisis management) where its overnight. IE she goes to her work from like 12 AM to 8 PM. During on-call she will submit for mileage/use which her work pays out.

Her work has "work cars" for her to use and I have started having her check out one of those if she is on call which is sorta a hassle vs just using her own car.

She thinks I am paranoid and that insurance wouldn't have an issue if she was in an accident with our personal car. I think they would, but too afraid to ask my agent.

Any one can give insight? Thanks.