One possibility is that they become involved in the fight against dark magic in America. Since they have experience from their time in Britain, they could offer their skills and knowledge. They may also explore the different magical cultures within America, from the Native American magic to the magic practiced in big cities like New York. Their move could be a chance for them to expand their understanding of the magical world on a global scale.
He could have different interactions with the Hogwarts teachers. Teachers like Professor Snape might treat him (as Hermione) with a bit more respect due to Hermione's academic achievements. Harry could also use this opportunity to learn more about the inner workings of Hogwarts from the perspective of a different student. He might discover new secrets in the library or during private study sessions with professors that he wouldn't have as Harry.
One new adventure could be that they discover a family heirloom together. Since they are siblings, they might start looking into their family history more deeply. Maybe there's a hidden magical object that only their family line can access, and they go on a quest to find it.
One possible plotline could be that they leave to escape the chaos in the wizarding world of Britain. Maybe they are seeking a fresh start in America where they can explore new magical cultures. For example, they could be interested in learning about the different types of magic practiced by American wizards and witches, like the use of wands that are made from different materials compared to those in Britain.
One possible plot could be that they move to America to start a new life away from the wizarding war in Britain. They might discover new magical communities in America, like Ilvermorny, and get involved in protecting its secrets. Maybe they meet new friends and enemies along the way and use their combined skills from Hogwarts to face new challenges in this different magical environment.
It could reveal new adventures in the Forbidden Forest. Hermione might take the lead in exploring the forest, using her knowledge to navigate the dangerous areas. She could encounter new magical creatures and try to study them while also ensuring her and her friends' safety.
A new adventure could be related to a family - owned magical property. It could be a house or a piece of land that has been enchanted and is now in danger. Harry and Hermione, as siblings, would feel responsible for protecting it. They would need to figure out the source of the danger, whether it's a dark magic curse or a group of greedy wizards trying to take it over. This would lead to a series of exciting events as they use their combined wits and magic to save their family's property.
A really interesting one is 'USA - A New Chapter for Harry and Hermione'. In this, they move to the USA seeking anonymity. They choose a big city like New York. Hermione starts working at a magical library, which has a vast collection of books from all over the world, including some rare American magical tomes. Harry joins a group of wizards who are dedicated to protecting the city from dark magic threats, which are a bit different from the ones in Britain. Their relationship also evolves in new ways as they face these new experiences together.
Perhaps they get involved in the American wizarding politics. The political situation in the American wizarding world might be different from that in the UK. Harry and Hermione, with their experience from the fight against Voldemort, could contribute to solving some problems in the American wizarding community, like issues related to the secrecy of magic in a more urbanized and diverse American society.
They could meet new magical students at Ilvermorny. For example, a young and talented witch or wizard who is curious about their experiences at Hogwarts.
One possible adventure for Hermione as a Time Lord would be to explore the origins of magic itself. She could travel to the very beginning of the universe, where time and magic first intertwined. There, she might discover the true source of all magic, which could change the way magic is understood and practiced in the wizarding world. She would also face challenges like dealing with powerful time - based entities that don't want her to uncover these secrets.