After Mog rejoins you, you can make a new friend in Narshe.

Go Toward the Esper

Go back to Narshe, which is in the middle of the north part of the world map. In Narshe, go north up the middle of town, and go into the cave where you first found the frozen esper. When the path splits, go east. Keep following the straightforward path until you reach the area where you prevented Kefka's troops from reaching the Esper. In this area, unequip the Moogle Charm/Molulu's Charm if it's equipped so that Mog can be in a battle and learn a new Dance.

Ice Dragon

Ice Dragon is weak to Fire.

Tritoch/Valigarmanda

Go north from the Ice Dragon's area and follow the path until you reach the frozen Esper. It is weak to Fire. It can't be harmed with physical attacks except with weapons like Drill and the AtmaWeapon/AtmaWepn/Ultima Weapon. It will attack with Ice 3/Blizzaga.

After the Esper

When the Esper is defeated, you will get the Tritoch/Valigarmanda Esper, and then a piece of the cliff will fall. Put Mog in your party. Then walk north to fall off of the cliff from that piece that fell, and you will end up in Umaro's Cave.

Umaro's Cave

Make sure that Mog is with you. After you land in the cave, go through the middle of the three doors. Go north and west and open the chest. Then ignore the other chest and instead go back the way you came, to the previous room.

Go through the door on the left, but don't step on the two darker spots on the floor, or you will fall through to the floor below. In the next room, open the chest to fight a Monster-in-a-box, Pugs/Tonberries. If you defeat them, you win a Minerva/Minerva Bustier.

Now go to the lower floor, either by falling through a hole in the floor or by going through the upper door in the room with the Pugs/Tonberries chest.

In this lower floor, there is a chest in the southwest part of the room. Use the short stairs in the middle of the area and cross the bridges to reach that chest. Then go east along the bridge, and then go north on the long bridge. Go up the stairs in the northern part of this area.

Go east and ignore the first switch, then go east and flip the second switch. Go north and check the skull to get an Esper. Then Umaro will appear, and you will have to battle him.

Umaro

Umaro is weak to Fire.

After you defeat him, talk to him, and if Mog is with you, Umaro will join you. You can use warp to leave the cave, or you can go up the stairs to fall out of the cave.