I'm probably not qualified to answer your question, but I will anyway. As such, take it for what it is: free perspective from an anonymous source over the internet.
Seems to me that biglaw isn't the place you "go to". It's the place you "come from". Although I have no legal experience, I do have a lot of corporate experience and there are some companies like this. Some consulting firms, P&G, IBM, General Electric, etc.
Once you've taken that first job elsewhere, it's difficult to break into those companies.
I did know a couple of biglaw associates and they both said that not only did their firm have a few people who graduated from lesser schools (Cleveland State was the one they mentioned specifically... I didn't debrief them, so there may have been more), they also said that there were people who transferred into the firm after having a career elsewhere.
I will only say this: to get in, it appeared that they had already established careers that were comparable to the duties they were going to perform in biglaw.
With Jones Day, they have links for "experienced candidates" to click on. Of course, this is a big firm (that's an understatement) and with people constantly leaving in order to raise families, etc., they have a big need for associates.http://www.jonesdaycareers.com/opportunities/
I guess I don't want to discourage you, but it's entirely possible that this boat has already sailed. There are just some things you can't make up for later.
If you want to be a supreme court justice and you didn't go to Harvard or Yale (probably Yale), or maybe Stanford (if you're a diversity pick), you just don't stand a chance. All the work in the world won't help you.
If you want to be a Law professor and didn't graduate from a T14, and probably more like the top half of those schools, it's just going to be darned near impossible. (And I swear, half the law professors in the country graduated from Harvard or Yale, so, your odds diminish as you go down the food chain.)
Same thing with biglaw: the time to do it is right out of school. Is it impossible otherwise? No, but for most intents and purposes, it's exceedingly difficult.
I doubt you'll get much guidance on this because the people who have done it are very few and far between and probably had a good measure of luck involved.