Itinerary help

hey! traveling to Japan in mid November for 13 nights and need some insights on a few nights:

Tokyo – First 3 nights

Hakone – 1 night

Kyoto – 4 nights (osaka day trip)

3 nights open – thinking Kanozawa for a night or two and then somewhere else?

Last 2 nights in Tokyo to finish…

Any suggestions on where to spend the 3 nights in between kyoto and Tokyo? Thinking somewhere authentic / off the beaten path a little bit? We are from New York, in early 30's. Thank you!