After Dr. Andonuts completes the Phase Distorter III, you are sent to Giygas's location in the past.
Go to Giygas
Do not save the game in the past. If you do, you will not be able to explore later.
Go west from the Phase Distorter. The path through the cave is straightforward, but full of dangerous monsters. Open the gift box along the way to get the Legendary Bat. When you find large metallic orbs, touch them to be transported to the next area.
There are a couple of Magic Butterfly areas in the cave. One is next to a hole in the floor, and the other is after the first metallic orb.
After the second metallic orb, go north to find a cave. Go inside. Walk north on the machine. At the top, you find Giygas.