Some of the onsen have hourly private rooms. For example, Hakone Yuryo in Hakone has shuttle just across the street from the Hakone-Yumoto train station at the bus station. You can go to the public shared onsen, or see if they have a private available. They take reservations up to 30 days in advance as well.

http://www.hakoneyuryo.jp

Some other onsen have the same, this is one I know for sure.