Shirahone onsen is great and they're famous for nigoriyu (milky-colored onsen), but it'd probably be hard to get there without a car. I can't remember what time of the year I went there, but I don't remember it being suboptimal temperature-wise. If you're an onsen snob who craves high quality of the onsen water, then it's worth putting Shirahone on your list imho. But I'm not sure if there's the type of ryokan there that would suit your preference.

There's Asama Onsen near Matsumoto, which I've read has an actual onsen town ambiance. But I've never been there and I don't know that you can get there without a car.

The one place near Matsumoto that you can get to just by train is Kami-Suwa Onsen. There's no real onsen town there per se. However, some of the hotel rooms overlook Suwa Lake and it's a treat to soak in onsen while enjoying the lake view. I like the ryokan Shinyu there. It may meet your criteria of a modern ryokan... it's newish and chic with very good food. But you'd probably want to get one of the rooms with en-suite onsen, because their public bath isn't much.