With Hades defeated, you now have the ability to swim through the River Styx without losing any health or magic. Dive down into the water and press the Square button. You will be swimming through the dead body of Hades. Once you see the surface, swim up.
Soul Summon will finally be unlocked thanks to the power from the Claws of Hades. With the Soul Summon, you can call forth the spirits of the cursed souls. Test it out on the enemies in this area.
Continue swimming and pull the lever to go back to the Forge. Back in the Forge, you will have another cut-scene with Hephasteus. You will now learn more about why he was banished in Olympus and her daughter Pandora. With the Soul of the Hades in your hands, you just opened a Hyperion Gate to pass through.
Go back to the place where you started. From there, you will see that a Hyperion Gate has been unlocked for you. Save first before proceeding and you will see Athena before going through the gate.