I live in a small city as well, so I can relate with your situation.
First thing I would do is to sit down with Google Maps and pinpoint the tourism and shopping locations both in Matsue and the vecinities. The local government probably has this info on its website, but there are probably places that are not covered, so check with your local friends. These places are where people are flocking and events are happening. Decent places for dates too. After that, do a small research on transportation both inside the city and the nearby towns. If there's some nightlife, is going to be probably close to a station: look up the izakayas, love hotels, taxi and bus stops, karaoke, etc. If Matsue is the biggest city in the area, people from nearby small towns are going there too. Then, check the nearest big city and see how far you are: that place can be useful both for chatting girls or inviting a date along with you. Matsue's young people are in the same situation as you, so they are likely going there during weekends. Finally, think about your own logistics: where do you inside the city, can you drive, where the trains and buses are, etc.
With that info at hand, you can start thinking what to do. You will probably have to get a bit creative though. Events, matsuri, cultural/language exchanges and community gatherings can also be opportunities to meet new people. I wouldn't discard the dating apps, keep them as another method.
I don't know yout Japanese skills, but having a decent level would really help in the countryside.
Take it step by step: small cities are harder than Tokyo or similar cities (fewer people, bad transport, fewer venues, small nightlife, good English speakers are rare) but it can be done.