(Disclaimer: the image above is from Shibu Onsen in Nagano, not far away)
As an alternative to heading up to Daisetsuzan after a few days' recovery in Matsumoto, we could instead make our way through neighbouring Gunma prefecture and Nikko before heading back to Tokyo to fly home.
This option would be easier, more relaxing, rich in Japanese history, and give us more time to enjoy just being in a place instead of being on planes and buses etc.
We would pick up a rental car in Matsumoto, drive through to Joshinetsu-Kogen National Park in Gunma for 2 nights, then to Nikko for 1-2 nights where we would drop off the rental car. From Nikko, we'd catch a train back to Tokyo.
The drive from Matsumoto to Gunma would be about 3hr 15min. The drive from Gunma through to Nikko would take about 2hr 30min.
Gunma
Gunma is a mountainous region most famous for its onsen, including Hoshi Onsen Chojukan, one of the best baths in Japan according to one of the onsen books I own (I have two).
I am picturing us staying 2 nights at Hoshi Onsen Chojukan, which is in Joshinetsu-Kogen National Park, with a day of hiking in between.
Hoshi Onsen Chojukan is a 140 year old Meiji-era ryokan, and the vaulted wooden ceiling above the Hoshi No Yu baths is supposed to be particularly impressive. Staying here would be a bit of a luxury, but a wonderful way to experience traditional Japanese hospitality.
Takumi no sato
After leaving Hoshi Onsen Chojukan we could stop at Takumi no Sato, an arts and crafts village about 20 minutes away by car. There are over 20 workshops in the small village, including soba, washi paper, lacquer, indigo dyeing, bamboo weaving and woodwork. This could be a great place to find some small, handmade souvenirs.
Then, onwards to Nikko.
Nikko
Nikko was apparently established about 1200 years ago when a Buddhist monk founded what is now known as Nikkosan Rinnoji Temple. It then became a favourite area for shoguns to build ornate monuments to honour 'kami' (Shinto gods), Buddhist deities and themselves. I have no evidence to support this, but I imagine they were inspired to build here because it is a beautiful area. Part of Nikko is now a UNESCO World Heritage site - specifically, two Shinto shrines and one Buddhist temple (together comprising 103 religious buildings) "located in an outstanding natural setting". The UNESCO listing details for Nikko are here.
Nikko is also set on a crater lake and is surrounded by natural beauty, with abundant mountains, waterfalls, lakes, gorges and marshes in the neighbouring Nikko National Park.
Nikko can be visited as a day trip from Tokyo and will always be busy, but it isn't an easy day trip and I think if it isn't a weekend then crowds will be manageable. I imagine it would be very easy to spend 2 nights/1 full day here.
It looks really gorgeous, but I actually don't know anyone who is been there. Most everyone heads down south to Kyoto instead. See here, here and here for some online resources.
From Nikko, it is just over 2hr on the train to get back to Tokyo.