Not a whole lot of public land that I know of in this part of the state. Might find some out in Jo Daviess County. There are some state parks that allow bow hunting but you have to check with each individual one for their rules and regulations.
to my knowledge, most all of the core of engineers land along the rivers is open to hunting. I did quite a bit of hunting in IL in those locations in years past. Also, some of the parks have sections open to daily sign in hunting for non-res. I've gotten into some really good hunting on a couple of those, too, so long as you are willing to go a smidge deeper than the locals. The catch is that a significant portion of their parks lands is either a draw or open to residents only, though. So, be sure to check that.