No, your fleet policy will not offer coverage for you when driving any vehicle. This type of auto insurance policy, like almost every other type of auto insurance policy, is tied to the vehicles, not the drivers. This means that you’ll only be covered if you’re driving one of the vehicles covered by the policy.
It won’t cover you if you happen to drive another company car that isn’t covered, your own personal car, the car of a friend or family member, or any other vehicle that isn’t a part of your fleet policy. You’ll need to take out additional policies to drive other cars, or you can become a listed named driver on someone else’s policy in order to drive their car.