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.
Obviously, the Primordial of Monsters itself could be the main antagonist. It has the power and the motive to cause chaos and destruction, which would put it directly in opposition to Percy and his friends.
It could be a new, powerful Titan that has emerged from the depths of Tartarus. This Titan might be jealous of the eldest primordial's power and wants to either destroy it or take its power for itself. And since Percy is associated with the eldest primordial, he becomes the Titan's target as well.
Another antagonist could be a primordial from another realm who is jealous of the power of the primordial of time. This primordial decides to invade the 'Percy Jackson' world and try to absorb the power of the primordial of time. It would bring a whole new level of threat as it has its own set of powers and minions from its own realm. It could also be that this primordial has the ability to twist time in its own realm, and when it enters the 'Percy Jackson' world, it starts causing chaos with time there as well.
It could be a power - hungry Atlantean noble who wants to rule not just Atlantis but also the surface world. He sees Percy as a threat to his plans and tries to stop him at every turn.
A new and powerful demon that has emerged from the darkest corners of the underworld. This demon is bent on using Percy's connection to the overworld to break through the barriers between the two worlds and unleash chaos. It has powers that are different from what Percy has faced before, like the ability to manipulate the very essence of the underworld's environment.
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 it could be a demigod who has become obsessed with power through gaming. This demigod, let's say a child of Hades, discovers a way to use dark magic and combine it with gaming technology to try and rule over all the other demigods. He sees Percy as a major threat to his plans because Percy is so popular and has strong powers, so he constantly tries to undermine Percy at every turn.
The goddess Hera might also be a great antagonist. Since she has a history of interfering in the lives of demigods, she could be jealous or threatened by Percy's rebirth. She might try to stop him from reaching his full potential in his new life or cause trouble between him and his new allies.
An ancient and forgotten god who has been awakened. This god could be angry at the modern demigods for ignoring the old ways and decides to send assassins after Percy as a warning to the rest. They might want to bring back the old order where gods were feared and worshiped more and see Percy as a symbol of the new, more independent demigods.
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.