Vetala is just at the bottom of the stairs. See here for details. Your reward is a piece of Cobaltite . A Librascope can also be found here.
After you’re farmed to your heart’s content, travel down the corridor to the next area where another Menhirrim stands watch and a Tyrant lurks. Kill it first and then talk to the Menhirrim to receive mark Mission 26. Proceed to the outer ring and approach another Mehirrim for a cutscene where that pesky fal’Cie blocks your path. Return to the sixth floor by heading back up the staircase (near the Pulsework Gladiator battle).