Best I could tell you close to Carbondale, would either be areas near Ava, IL, Kinkaid Lake, Cedar Lake (near Pomona, IL), La Rue Pines area, Giant City State Park, and of course the Crab Orchard Nat'l refuge. Just matters how far you wish to drive.
No Permit required to my knowledge for bowhunting at least, state parks require sign in and sometimes special tags.
Very easy to hunt especially since we have had no rain in over a Month till today. Water holes are working well and the Persimmons are falling now. I have hunted the refuge but not much other than that for public land. Plenty of big deer on the refuge.
Refuge page