There is a lot back and forth, but another important augment just spawn and is located in Lodestone Cavern. Make your way there, ride a black Chocobo, get into the cavern, pick up the Whirlwind Augment, and get out. You can use the same trick to fast exit the cavern by using Teleport outside the Crystal Chamber.

Your next destination is Agart Town which is located in the far south on a small island. You have already been there once during item collection.