There was a thread on this site in early October regarding hunting at Lake Hope State Park and the Zaleski State Forest. You might do a search and see a lot more detail about some of what is happening in Ohio.
For a variety of reasons the deer population has dramatically decreased over the past 15-20 years. Vinton county, where most of the Zaleski state forest resides, used to be one the top three counties for big deer. No more. Private land around the forest can be pretty good, but access is almost exclusively by hunting lease.
Some of the best hunting now is around urban areas where deer populations can grow almost unchecked. A number of large cities with expansive parks have begun limited access hunting, often archery only, to help control the herd. Getting access can be tough. Often requires things like police backround checks, proficiency testing, lottery draw, etc. These programs are almost never publicized. Word of mouth and 'friend of a friend' is how most folks find out about them. A way to find out about them is to inquire at the local park board, or the police station. Folks will answer your questions, but seldom volunteer any information. Around where I live, about 10% of the folks who apply actually get to hunt. So even if you find a program, there's limited chance you can participate. No one, to my knowledge, has implemented any kind of point program to help enhance your chances based on the number of years you apply. So you start over each year.
But the hunting in these areas is not for me. The concept of hunting out of a blind, or tree stand overlooking someone's pool, is not appealing to me, but there is something to be said for hunting where the deer are.