Most shinkansen trains beyond Osaka that you can use your JR pass on don't have green cars to begin with (eg. Hikari Rail star or Kodama going to Himeji, Hiroshima) Miyajima from Hiroshima is a local train no green car.

Are you sure? I think the Hikari service which stops at both Himeji and Hiroshima has green cars unless it has changed in the last few years since I travelled from Narita to Hakata in Green Cars.