We believe you are referring to underinsured motorist conversion coverage. This type of insurance is available in Connecticut.
Connecticut's Insurance Department notes that underinsured motorist conversion coverage provides for reimbursement in case you are injured by an underinsured driver. If your damages exceed the at-fault driver's insurance, or other payments, your underinsured motorist conversion coverage will be available for damages not paid. This optional coverage is not reduced by payments from any other source, including the at-fault driver's liability insurance.
For example, if you have uninsured/underinsured coverage of $300,000 and you sustain $400,000 in personal injuries caused by an at-fault driver with $200,000 in bodily injury liability, without underinsured conversion coverage you would normally only be able to collect the $200,000 from the other driver's insurance plus $100,000 from your own underinsured coverage, which equals your $300,000 limit. With conversion coverage in CT, you could collect the full $200,000 from your own underinsured coverage, thus getting a total of $400,000 instead of $300,000.
So in Connecticut conversion coverage is an option that may be purchased which, in effect, "stacks" your policy onto that of a tortfeasor (at fault party). In other words, if someone at fault has insurance coverage that is not adequate to pay damages for injuries which you sustain, conversion coverage allows you to place your underinsured motorist coverage on top of the responsible party's coverage regardless of what your limit of underinsured motorist coverage may be. This type of stacking is available in other states but the term "conversion coverage" is unique to Connecticut to the best of our knowledge.
This "conversion" underinsured motorist coverage is in contrast to the typical scenario whereby you would only be able to collect that amount of underinsured motorist coverage which is in excess of the other party's policy. So with a normal underinsured motorist bodily injury (UNDUM) policy if you had $50,000 dollars in underinsured motorist coverage then your total compensation from both policies would be $50,000.
We do not have a comprehensive list of those states that have what Connecticut terms to be conversion coverage. If you are interested in which states may allows this conversion or "stacking" of underinsured motorist coverage you can contact their insurance regulator for a list of required and optional auto insurance coverages.