If you feel like taking a one day plan, you can rent a bike, actually is kind of borrowing a bike cause is totally free, then go to Yoyama park enjoy the amazing view of the city (it could be a bit too hard to go up, but it is worthy). Then go straight ahead to Osama onsen (only about 30 minutes by byke)
If don’t feel sporty today, just enjoy the benefits of Ryushu onsen, which is just about 15 minutes away from Matsumoto Backpackers hostel (by walking).
Just keep in mind that after the extremely relax sensation, you would probably just want to go to bed, so it is better to go late at night, as the last entrance is at 22:30.