As of the date of this post the mandatory minimum legal limits of insurance coverage to drive an automobile in the state if Illinois are $20,000. An effort was made this year to increase the liability limits, but it was killed almost immediately.
I estimate that about half of all drivers with insurance in Chicago only have $20,000 of insurance coverage. Think about this... One night in the hospital could cost way in excess of $20,000.
Most people in Illinois aren't overly worried about under-insured motorists becuase most people in Illinois carry under-insured motorist insurance coverage as part of their regular bundle of auto insurance. Car-free cyclists, on the other hand, typically do not have auto coverage, so if they are involved in an accident with a negligent motorist the only insurance is the motorist's.
I suggest that bicyclists carry their own auto coverage for only the under-insured/uninsured motorist coverage. An operator's policy costs less than a couple hundred dollars a year, and it can offer a good deal of coverage for under-insured motorists, uninsured motorists, and hit-and-run motorists.
Safeco and Progressive both offer such coverage as "operator's policies" or non-owner policies. It's cheaper than regular auto coverage becuase you don't own a car. I know car free cyclists who sometimes rent a car. If they have a non-owner's policy they can forgo the rental insurance. If you rent a car a few times a year, the money you save in optional insurance will pay for the non-owner's policy, and it will cover you all year round.
If you have questions about this type of coverage please feel free to call to talk about it.