Getting a non-resident permit from another state like Utah or Florida would allow you to carry in that state and any state that recognizes that states permit. Some state only recognize out-of-state permits if the person with it is a resident the state that issues it. Since Illinois honors no out-of-state permits, having a Utah or Florida non-resident permit would do you no good in Illinois.
Someone told me I'd have to either become an alderman, get deputized by my CLEO, or become a PI or something...
You would need to look at Illinois state code to see what exceptions exist for concealed carry, but I expect they are rare. The NRA-ILA page shows no concealed carry at all. Open carry is allowed for some activities like shooting on a range or hunting, or on your own property.