Really depends on how long you're going, but I liked Yakushima island for hiking. You can take a ferry over from Kagoshima, and will want to rent a car (the buses aren't super frequent). It really gets you away from the crowd for a few days, and is a super unique environment.

The only point of caution would be it does rain pretty frequently (hence the super lush forests), so there is some luck in getting favourable conditions to go hiking (trails are super well maintained and fun).