There are many facts missing from your question, but as a general rule, you can appeal an adverse decision from the Ohio Supreme Court to the United States Supreme Court if there is an issue concerning the U.S. Constitution or a federal statute.
Filing a motion for a new trial long after the verdict would usually require the recent discovery of new evidence.
No matter which road you pursue, finding an experienced appellate attorney as soon as possible should be your first step.
I'm pretty sure I addressed this question in another form elsewhere, but basically, if you have already run your direct appeals out, the next stage would be seeking a writ of habeas corpus. You can do this by bringing a 2254 action in federal court for any issues you raised in the appeal and were addressed on their merit which affect a federal constitutional right or you may (depending on time and some other factors) be able to go through the state habeas process first and then possibly still get to a 2254 if that does not work out.
The key, and I mentioned this elsewhere, is that you have to get with an experienced post-conviction attorney pronto. Post-conviction actions are civil in nature usually and that means they deal with very strict time deadlines which will leave the defendant totally out of luck if they are missed.
You should consult with an appellate attorney in Ohio to see what else, if anything, can be done.
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