For most travelers, the trip ends with a Shinkansen ride back to the airport, and perhaps a box of Pocky from the snack cart. If you're going to Tokyo, don't forget to look out the window for Mt. Fuji, on the left-hand side of the train as you're passing through Shizuoka. But if you have more time, there's much more to see in every direction.
If you're traveling east from Hiroshima or Okayama, the city of Himeji is right over the border of Hyogo prefecture. There's really no overstating the awesomeness of Himeji-jo, the granddaddy of Japanese castles, though parts will be under renovation until spring 2015. It's a short walk down the main street of the city from the train station.
If you enjoyed Izumo Taisha, the second holiest Shinto shrine, then a trip to Ise is definitely in order — the forested Ise Shrine（伊勢神宮) is the holiest of all, split into Gekū (外宮), the Outer Shrine, and Naikū (内宮), the Inner Shrine. At Osaka, switch to the private Kintetsu rail line (1h40m, ¥3030).
The San'yo Shinkansen terminates at Hakata in Fukuoka, which is two stops on from Shimonoseki, over the Kanmon Straits. If you're hungry, sample some of Hakata's much-touted ramen at the station. Travelers with an interest in the atomic bomb may wish to step out at Kokura for a moment and observe the incredible power of chance — that was the original target of the bomb that, due to a cloudy day, was dropped on Nagasaki instead.
From Kokura or Hakata, connections can be made via the Sonic Express to the goofy jigoku (hells) and hot springs of Beppu. The Kamome (かもめ) Limited Express runs from Hakata to Nagasaki in about two hours.