Maybe join the NWTF (National Wild Turkey Federation). There's a chapter in Yorkville. I don't know anything about it other than it exists.
Here's the list of left over permits:
http://www.dnr.illinois.gov/hunting/...y01DEC2011.pdf
Go to the dnr website and download a paper application, it will show the abbreviations the dnr uses for different public sites. If you have access to private land anywhere, it might be a different story, but otherwise, pickings are pretty slim at this point. If you are hunting with kids, you may be able to get youth tags as well, which would improve your options (deer or turkey).