After leaving Mine-ji I passed through the town of Mitoya on my way upriver to the next pilgrimage temple, Zenjoji, and stopped in at what appeared to be the main shrine of the town, the Tenmangu.
The newly reconstructed zuijinmon had curtains blocking off the zuijin from view. Being a Tenmangu, the main enshrined deity is Tenjin, considered to be the spirit of Sugawara Michizane.
Curiously it lists Amaterasu along with Tenjin as the main kami, though I strongly suspect that she is a fairly modern addition.
According to the signboard Michizane spent a few days here during the reign of the Daigo Emperor, which was the time Michizane was "exiled" to Dazaifu. His route to Kyushu from Kyoto would have been through the Inland Sea, so its not clear to me why he would make a trip over the mountains to Izumo. A few hundred meters from this shrine is a small Sugawara Shrine which would seem to reinforce the story.
There are several smaller shrines within the grounds but the only one I could identify was the Inari shrine.