Looks like there are some nice Public Hunting access in your area:

Kaskaskia River FWA is 34 miles from FairView Heights and it's 14,000 acres and it's looks like the largest public hunting area in your region.

Horseshoe Lake SP has 4195 acres divided between 2 refuges that are both bow only and they are about 19 miles from FairView Heights.

Peabody River is a 1,000 acres and it's bow only and it's 35 miles from you.

I hope this helps. I got all this info off of the DNR website. Hers is the link. Just go down and click on Public Hunting areas toward the bottom.

http://dnr.state.il.us/admin/systems/index.htm