Hi there,
Here are a couple of example routes. These are all fully covered by the JR Pass.
The problem with this route is that Hagi is very remote and there are not also good transfers. The full route does indeed take around 10 hours and there are only a few options per day.
Not shows in Hyperdia is the JR Chugoku but between Shin-Yamaguchi station and Hagi. You can also use this bus for free using the JR Pass. You can find the time table here: http://www.chugoku-jrbus.co.jp/pdf/route_bus/yamaguchi/20161001time_yamaguchi_3.pdf
It is only available in Japanese, so here is a little explanation for it.
The top time tables are for normal week days.
The time table on the top left is Shin-Yamaguchi to Hagi, the first stop is Shin-Yamaguchi and the last Hagi.
The top right is for Hagi to Shin-Yamaguchi. Hagi is the first stop and Shin-Yamaguchi the last.
The time tables at the bottom are the same as above but for weekend days and national holidays.
You can plan your trip from Shin-Yamaguchi to Tokyo using Hyperdia.
Hope this helps,
Daniel