I live in a fault-based, or “tort liability" state. I have all the usual auto insurance, including $2000 per person medical payments and $50000 bodily injury liability.

If a passenger in my vehicle is injured in an accident that is my fault, and that passenger does not have health/medical insurance, and the hospital bill for that passenger's injuries exceed $2,000,- then I presume that passenger would want to sue me to recover the difference between their actual medical costs and the $2000 my insurance paid.

What kind of insurance would cover me if I were sued under these circumstances? Should the bodily injury liability insurance kick-in? My agent says no,-- that is for passengers in other cars, not in my car.

But it seems to me I should have some kind of insurance to cover this scenario. I mean after all, if that's the case, I should never carry a passenger unless I first verify they have adequate medical coverage in case of an accident?
Additional Details
As a follow up to my question- in this case the passenger is a family member, but not a family member that lives in my house. Should that make a difference?