This page lists the things that you can do in Yo-Kai Watch 2 after completing the main game.
Rank S Watch
After completing the main game, you can talk to Mr. Goodsight at Timers & More to get the quest to upgrade your watch to Rank S.
The Infinite Inferno
Visit the Stone House in the northernmost part of Breezy Hills and talk to everyone there to get the Shack Key. Then go to Blossom Heights and go west from Shoten Temple to find Prayers Peak Tunnel. At the end of the tunnel, you find the shack. Unlock it and go down the ladder to find the Infinite Inferno. You can fight rare enemies and tough bosses here.
The Divine Paradise
You will reach the Divine Paradise after you defeat all of the bosses of the Infinite Inferno. The Divine Paradise is another place to find rare Yo-kai.