I believe you're going to Osaka on day5 and returning to Tokyo on day9.
Are you flying out of NRT on day11?
If so, and you're getting the 7-day JR Pass, then activate it to start on your day5 and end on day11 so that your bullet train and return NEx are covered.
On the other hand, if you're going far enough via JR for your day hike trip from Tokyo like on day 3 or 4, then you'd want to have your JR Pass active between days 3-9.
You could do a day trip to Miyajima from Osaka and do some easy hikes on Mt.Misen where the inland sea views are outstanding from the top.