Hi!
Completely off topic, but for anyone interested in the bat saga, our bat has flown off to better abodes and fortunately seemed to have been healthy after all, much to my relief. Oddly this does connect to Hakone because I believe the ryokan @Lazy Pious likes has a bat theme based on their webpage.
Going back to OP. While this is not somewhere I would suggest to you (based on both cost and because it's not everyone's cup of tea) I have personally done a 5 hour round trip from Tokyo in November. (Licca Castle in Ononimachi, Fukushima-ken.) A lot depends on if you care about how late you get back. So going to Matsumoto I think only would add another hour to your roundtrip as long as you don't mind getting back when it is dark, it's possible or you could add an overnight in Matsumoto since hotels tend to be quite affordable there. Matsumoto is a relatively cheap city.
Or knock a night or two off Tokyo and look into going to Atami and the Izu Peninsula as well if you don't want to try Matsumoto. You could even take all local trains and cut the trip shorter by combining it with the trip to Kamakura. At Ofuna it's easy to change to trains heading to Odawara and Atami. One of my trips to Hakone I did that. Ofuna also has trains to Narita Airport.
You could also look into overnighting in Nikko and getting one of the Tobu passes where you can travel all the way up to Aizu Wakamatsu. You'll just need to come back to Nikko to leave or look into a bus, because JR from Aizu Wakamatsu back to Tokyo gets pricey because it's ~8000 yen for the shinkansen from Koriyama to Tokyo.
I think Chichubu could be really interesting. It's one of those places I keep meaning to hit up. Mitsumine jinja looks really cool and thanks to the link @JapanCustomTours gave earlier in the thread, I was reading about the search for the possibly not extinct Japanese wolf in the area via the Japan Times.
I seriously respect @Lazy Pious' opinion, but I think if you're on a tighter budget, Hakone is really sort of expensive. It's going to be a 620 yen round trip no matter what to get from Odawara to Hakone-Yumoto.
If you want to go all the way to Gora that's another 400 yen unless you bought the ticket with your fare to Hakone-Yumoto which drops it to 670 one way from Odawara to Gora. It's 360 yen to take the pirate boat that 5-10 minute trip from Hakone-machi to Moto-Hakone (
https://www.hakone-kankosen.co.jp/foreign/en/timesheet/) Once you start adding in a bus or two to get to Moto Hakone it can really add up as a day trip or overnight. It's one reason people tend to get the Hakone Free Pass.
I have different dates than you and maybe the recent volcanic activity will lower prices, but when I went in 2016 and when I was looking this trip and my July 2018 trip, everything was $200 USD a night or more. I know I once stayed somewhere excessively cheap, but my friend found it on the Japanese version of either Rakuten or Jalan which do have different offerings than their English counterparts. The place also could have closed since we're talking over a decade ago.
I admit, for me, too much time in Tokyo == spending a lot of money because I tend to shop in Tokyo, but I know most people who post here are not shoppers lol
Good luck!