If you go outside Tokyo, you can encounter natures right away! Japan is filled with mountains, rivers, and oceans. Nature is everywhere and probably "secret" ones would be those next door natures.
I'm not confident that these are secret ones, but Tohoku regions has great nature. For instance, Oirage Gorge in Aomori Prefecture. I visited there in winter and participated in snow hike experience. I encountered to magical sceneries there. We went on a weekday and we met only 2 groups in the forest.
Yamadera temple in Yamagata Prefecture is also very beautiful. It is a temple built on top of a mountain, and you will have to hike over 1,000 steps to reach to the temple. The view was amazing from the top.
If you go south, Yakushima is astonishing. It is an island located in southern part of Japan, said to gave Hayao Miyazaki to create the famous Ghibli film Princess Mononoke. The forest was so beautiful and I was surprised to find such a different scenery in same country Japan.
You should definitely experience a hike to Jomon-Sugi.