He could run to the Forbidden Forest. It's a place full of mystery and danger, but also a good hiding place as it's vast and not many people would expect him to be there. He might find some solace among the magical creatures there.
Harry might run to the Room of Requirement. It could transform into a safe haven for him, providing him with everything he needs to hide and stay safe from Snape. It's a place that can only be found by those who truly need it, so it would be a great hiding spot.
One possible reason could be that Harry is afraid of Snape's intimidating presence. Snape has always been strict and rather unfriendly towards Harry in the original series, so in the fanfiction, this fear might be amplified, causing Harry to run. Another reason could be that Harry has stumbled upon something he shouldn't have seen that involves Snape, and he's running to avoid getting caught.
One reason could be fear. Snape has always been rather intimidating to Harry. Snape's cold demeanor and his past interactions with Harry's father might make Harry feel constantly on edge around him. So, running away could be an instinctual response to this fear.
Maybe Harry feels a sense of guilt. In the context of the war, he might think that he's failed in some way by getting into this unexpected situation. He could believe that his new condition might put others at risk or that he's no longer the hero everyone expects him to be. So, to protect others or to avoid the shame, he decides to run away.
Snape might suspect that there is some dark magic at play. Given his knowledge and expertise in the Dark Arts, he would immediately look into any signs of foul play. He could also be conflicted because on one hand, Harry is a student he has had many issues with, but on the other hand, he is also an important figure in the fight against Voldemort. So he would have to decide whether to report Harry's disappearance or try to handle it privately.
Snape would likely be angry at first. Harry has always been a source of trouble in his eyes, and running away would just add to that perception. He might mutter some sarcastic remarks under his breath about Potter's irresponsibility.
He could meet American witches and wizards. Maybe a young and adventurous American wizard who shows him around the American magical world.
Perhaps he runs away because he feels misunderstood. In the fanfiction, the other students or even some of the teachers might wrongly accuse him of something. He could be tired of constantly having to defend himself. Like if he was wrongly blamed for a serious incident at Hogwarts, and no one seemed to believe his innocence, he might just leave in frustration.
Perhaps he'll meet a mysterious old wizard who lives in a secluded temple in the mountains. This wizard could be a master of ancient Japanese magic and could teach Harry about the spiritual side of magic that is deeply ingrained in Japanese culture. Harry might also meet a group of young Japanese witches who are training to become magical healers. They could share their knowledge of Japanese magical plants and how to use them for healing, which would be very different from the healing methods Harry learned at Hogwarts.
Well, perhaps they run away to escape the chaos at Hogwarts. Maybe they've discovered a dark secret within the school that puts their lives in danger. So they decide to flee and find a hidden place in the Forbidden Forest to stay safe.
One possible plot could be that they run away to escape the dark forces that are taking over the wizarding world. They might be on a quest to find a powerful magical artifact that could turn the tide of the battle. Along the way, they face various challenges like dangerous creatures and spells.