I don't know about Illinois but here in Michigan you can hunt public land with good success other than the November fire arm deer season.
Many farmers will give permission to hunt coyotes, foxes and ground hogs readily. We have gained permission to hunt other things (deer not so much a state holiday thing) at these farms after the farmer got to know us and see we left fences and gates as we found them always park where the farmer said for us to and eliminated coyotes and ground hogs.

Al