Perhaps it's a powerful and ancient monster that has a vendetta against Percy. This monster is so strong that Percy can't defeat it easily, and it keeps tormenting him and those around him, leading Percy to consider death as an escape.
It could be a new, powerful and malevolent god. This god might be jealous of Percy's status as a hero and the favor he has from some of the Olympian gods. So, the new god devises a plan that makes Percy's life so miserable that he asks for death.
It could be an ancient and powerful Titan who escaped from Tartarus when the Doors of Death started to malfunction. This Titan wants to keep the doors open to bring back all the old enemies of the gods and take over the world again.
It could be a powerful demigod who has been corrupted by power. This demigod wants to be the sole champion of the arena and will stop at nothing to defeat Percy.
A powerful demigod who has been corrupted by the idea of primordial power could be the main antagonist. They might believe that by allying with a primordial or obtaining their power, they can become greater than the gods. This demigod would then oppose Percy, who stands for the protection of the current order.
A powerful Amazonian sorceress who has turned against her own kind and is now trying to use dark magic to take over the world. She could see Percy as an obstacle in her path and constantly try to defeat him.
Maybe a rogue demigod who blames Percy for something and wants to make his exile permanent. This demigod could have some dark powers or be in league with some evil forces. They would constantly try to thwart Percy's attempts to return or find a way to clear his name.
Maybe an ancient sea demon that has possessed a pirate captain. This possessed captain is now leading his crew to cause chaos everywhere. They are attacking other pirate ships and even some demigod - related places. Percy has to find a way to exorcise the demon from the captain and stop the crew's rampage. The demon is very cunning and uses the pirate's knowledge of the sea to its advantage.
The main antagonist could be Nyx. She is a powerful primordial goddess of the night, and Tartarus is her domain in a way. She might see Percy as an intruder and try to stop him from escaping or achieving his goals there.
It could also be a powerful sorceress who has her own motives for disrupting the demigod world. She might be seeking to gain control over the Olympian gods by using Percy as a pawn. She could use her magic to manipulate events and people around Percy, causing chaos and confusion.
Perhaps an ancient and dark deity who has been awakened due to some event related to Percy's exile. This deity wants to bring chaos to the world of gods and demigods, and Percy, even in exile, is the only one who can stop it. The deity would use all kinds of dark magic and minions to oppose Percy and his friends.
The main antagonist could be a primordial being that has been awakened. It sees the phoenix as a threat to its existence because the phoenix represents rebirth and renewal, which goes against the primordial being's nature of chaos and destruction. So, it tries to destroy the phoenix and Percy has to protect it, along with his friends, using all their skills and the unique powers they possess.