The obvious antagonist could be the Titan of Earth itself. Since it's a powerful being, it could be causing chaos and destruction as it tries to regain its former power and influence. Its very existence might be a threat to the balance of the world that Percy and his friends are trying to protect.
Another possible antagonist could be some of the more power - hungry Titans. They might see Percy as a rival or someone who could disrupt their plans. They could try to manipulate him or turn him to their side, and if he refuses, they would become his enemies. Also, there could be some ancient monsters that sense Percy's new power and want to destroy him before he becomes too powerful.
The traditional Titans could be antagonists. Since Percy has only half - Titan blood, they might see him as a threat or a tool to use against the gods. Kronos, if he were to somehow return or have a remnant of his power, would be a powerful foe. He could try to manipulate Percy's Titan side to turn him against the gods.
The Giants might also be antagonists. They could band together and target Percy's kingdom, believing that by destroying him and his rule, they can undermine the power of the gods further. They might use their brute strength and strategic thinking to attack Percy from different fronts.
Another option could be a group of rogue demigods who have turned to the dark side. They follow Percy to Forks, hoping to find some powerful artifacts there. They don't care about the vampires or werewolves and are willing to cause destruction to get what they want. Percy has to stop them while also dealing with the local supernatural creatures.
The Titans could be the main antagonists. Even though they have been defeated before, some of them might have found a way to return or their followers could be causing trouble. They see Artemis as a threat or a target for their revenge against the gods. Another possibility is a new group of powerful monsters that have banded together and are specifically targeting Artemis for her power and influence among the demigods.
Well, the main antagonists could be a group of rogue demigods who have formed a cult around Kronos. They believe that Percy, as the heir of Kronos, should lead them in overthrowing the current order of the demigod world. But Percy doesn't want any part of it, so they become his enemies. Another possible antagonist could be an ancient being that was once in league with Kronos and now sees Percy as a means to resurrect Kronos or continue his dark legacy. This being would stop at nothing to force Percy to embrace his role as the heir.
Perhaps some of the older, more traditional gods could be antagonists. They might see Percy as a threat because he's now the God of Dinosaurs and they fear the power he could gain from controlling these ancient and powerful creatures. For example, Zeus might be worried that Percy's new role could disrupt the existing power balance among the gods.
The Roman gods could be antagonists. Since Percy is a Greek demigod, they might not fully trust him as the King of Rome. They could try to undermine his rule, believing that he will favor the Greek side in any disputes or decisions regarding the demigod world.
Some of the more powerful demons in Hell itself could oppose Percy. They might be jealous of his sudden rise to power as the Prince of Hell. These demons could have their own agendas and territories in Hell that they don't want Percy interfering with. They could use their dark magic and legions of lesser demons to try and get rid of him.
Gaea herself would obviously be a main antagonist. She has the power to manipulate the earth and all its forces against Percy. Her children who are fully committed to her cause could also be major antagonists. They might have powers related to the earth and be very difficult for Percy to defeat.