I have not hunted Columbus, so I have to say no to your question, BUT Ohio is rich in Civil War history, especially Columbus. Just West of Columbus was/is Camp Chase. Although you can not hunt camp Chase, you can knock on doors asking permission to hunt property around Camp Chase and other Civil War sites. Google Civil War Columbus Ohio and you will get all kinds of info. Here is one on Camp Chase.
http://www.censusdiggins.com/prison_campchase.html
No famous battles in Illinois either but this is what I found this Spring hunting around a Civil War camp in Illinois in three trips to the area. The upper display is the Civil War finds and the belt buckle was not found there, but everything else was.
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