Iwami Ginzan Silver Mine and Izumo Taisha Question
 

We are driving from Hagi to Matsue and would like to see Iwami Ginzan Silver Mine and Izumo Taisha without backtracking from Matsue. Below is my thought on timing. At first glance this seems doable (but definitely rushed) but please let me know if you think I should allocate my time differently. Also, are there places to eat near the silver mines or in Omori Town? Thanks!

Leave Hagi by car at 8:15am
Arrive to parking lot at 11am - Walk through Omori Town and Mining Area, with quick lunch, 2.5 hours total
Leave by car at 1:30pm
Arrive at Izumo Taisha 2:30pm - visit for 90 minutes
Leave by car at 4pm
Arrive Matsue at 5pm

I have been to Izumo Taisha, but not the silver mines. I went from Matsue via train and back for the day. I can tell you that 90 minutes there is sufficient. It looks to me pretty worn down, but our Japanese exchange student told me it's the holiest temple in Japan, so I felt obligated to see it. Adjacent to the temple there is a long street with several restaurants and shops, including a Starbucks, so you might want to delay lunch until you get there.

There's a really nice museum near the Shrine - Shimane Museum of Ancient Izumo - but it's closed from November to April 2020.....
I agree with CaliforniaLady that 90 minutes is quite adequate to visit the Shrine.

Hagi is an often ignored yet very nice gem that few people see.
You may be understating the time it takes to get to the mine. The San-in side things go very slow - it might take you another half hour to get there, and even more if you choose regular non-toll roads. Also you didn't say where you are starting from, but if you can fit it in, seeing the Motonosumi Inari Shrine further west in Nagato is a real treat, plus north of Senzaki Stn are the Omijima Sea Alps, a very picturesque rugged coast and very nice, unless the weather stinks. It's unclear where you are starting or ending up, but for a car rental, the drop off charges might be quite high if you're not going back to Hagi.

I think you need all ninety minutes to visit Izumo Taisha. It is in a park setting, and the buildings are spread around a fairly large area. You don't want to rush it, and maybe you can take a few minutes to people watch in the cute town.

I hope I'm not throwing a wrench in your plans, but when I visited Izumo Taisha from Matsue via train, I took a little round trip bus trip to the Himonisaki Lighthouse, and back. The whole area was fascinating, and the coast is stunning, maybe a little like Point Reyes. I climbed to the top of the lighthouse, and what great views up there! I thought it was more interesting that Izumo Taisha. I think it might be en route for you, so you may want to make a substitution. Just my opinion, although I do have a thing for lighthouses.