That’s what I was leaning towards after my goggle search. But I found nothing with Illinois Special.
Any idea on the years I could have been produced?
I don't think there are any complete records for Crescent Arms, I heard/read someplace there were at least 100 known names and suspected at least that many more. Since any hardware store, merchantile, department store, auto parts store, cooper, hotel, or any other entity willing to buy a minimum number of guns could have whatever they wanted stamped on the gun, it means there are likely hundreds of "brand names" out there no one will ever know about. And that is just what Crescent is thought to have pumped out, there are several others that did the same thing, not to mention imports from Europe though that does not look like one.
As for date of manufacture, the roundish receiver looks to be 1900-mid 1920s. I had a Stevens very similar looking to yours and it was extrapolated to 1919 +/- a couple years. That style of receiver seemed to fade out in the mid-20s though I have seen records of some leaving the factory around WWII. Those may have been from parts misplaced and later found which was not unheard of back then.