Maybe a corrupted past version of a demigod. For example, a demigod who was promised great power by the Titans and turned against the Olympians. This demigod could see Percy as an obstacle to their plans and try to stop him at every turn. They would know the time period well and could use that to their advantage against Percy.
It could be Kronos. Maybe he's found a way to trap Percy in a time loop as part of his plan to stop Percy from fulfilling his destiny. Each time Percy breaks free from the loop a little, Kronos throws new challenges at him.
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 an ancient and powerful Titan who was thought to be defeated long ago. Since Percy is in the past, this Titan could be in his prime and see Percy as a threat to his plans of ruling over the world again. For example, Kronos might be even more powerful in this past era and Percy has to find a way to stop him without the help of his modern - day friends.
It could be an ancient being that was locked away for centuries but has now been released due to Percy's meddling with time and tides. This being might have a vendetta against the gods in general and sees Percy as a key to achieving its revenge. It could have the power to manipulate time in a different way than Percy, causing all sorts of chaos and destruction. For example, it could create time loops that trap people in never - ending cycles of pain and suffering, and Percy has to figure out how to break these loops and stop the being.
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.
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.
A new and powerful Titan could be the main antagonist. This Titan could be jealous of Percy's immortality and see it as something that should belong to Titans instead. He could gather a new army of monsters and try to overthrow the Olympian gods once again, with Percy being the biggest obstacle in his way.
A powerful nature - hating Titan who wants to plunge the world into eternal winter could be a great antagonist. He would be directly opposed to Demeter's power over the seasons and crops.
It could be a powerful demigod who was once rejected by the Camp Half - Blood. This demigod, filled with resentment, forms a mafia - like group to seek revenge on those who he believes wronged him. He uses his powers, which could be something like controlling shadows, to create an air of mystery and fear around his mafia operations.