(SPOILER)The other bonus dungeon appears once you beat the game, and make sure you wait until after the credits and 'the end' screen. It'll make a cut to the dungeon's location, so you'll know where it is.

The trials that open up are also available depending on who was in your party when you beat Zeromus. You'll have to beat the game a minimum of two, but more realistically three times to make all the trials available.