From the hot spring town Yugawara there is a small road twisting up the mountain towards Hakone. This is route 75, also known as the Tsubaki Line and has been a scenic route since a long time.
Starting from Yugawara, the best section is that from Yugawara until Mount Taikan, this is also where the Hakone Turnpike ends. This 13,5 kilometer section has an elevation of around 750 meters and has lots of corners with a few straight lines. Furthermore traffic is much less here than the latter part of the route 75. After Mount Taikan, the Tsubaki Line continues further to the cultural part of Hakone, which has a lot of museums worth visiting.
The road quality is generally very good but the the first half of the course is quite narrow, though climbing and descending traffic still have each there own lane. There are several viewpoints along the route but the one called Tsubakidai gives one of the best views and also is the entrance to a small waterfall deep in the mountains.
This route can be driven almost all year round. At the time of writing this route is toll free.
Google Maps location link:
Entrance to Tsubaki Line in Yugawara
JDM4all Youtube video: