Maybe they are following a lead on a new dark magic threat. If there is a new and dangerous form of dark magic emerging in America, given their track record of fighting against dark forces, they would be the ones to go and investigate. They would use their skills and intelligence to figure out what's going on and how to stop it.
They could be on a mission to exchange magical knowledge. Hogwarts has its own set of spells and magical techniques, and so does the American wizarding world. They might go to America to learn new things and share their knowledge as well.
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.
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.
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 reason could be to search for a particular magical item. Maybe Dumbledore left behind a clue that led them to believe there was something important in one of the vaults at Gringotts.
It might be related to Hermione's research. She may have read about some unique magical phenomenon in Australia that she wants to study first - hand. And Harry, being her loyal friend, goes along with her. For example, there could be a special type of plant with healing properties that only grows in certain parts of Australia.
They might encounter new magical creatures unique to America, like the Thunderbird. Harry could use his fame to make connections with American wizards, while Daphne might be fascinated by the different magical culture there.
They could meet American wizards who have a very different style of magic. Maybe some of the native American wizards who have their own ancient magical traditions. These wizards could teach Harry and Hermione about magic that is connected to nature in ways they've never seen before in Britain.
It could be for safety reasons. If there's a threat at Hogwarts or in the wizarding world in general, Hermione's house might be a safe haven. Her family might have some protective spells around the house that Harry can take advantage of. Also, Hermione's parents are Muggles, so it might be a place where the dark forces wouldn't expect them to hide.
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.
They could meet MACUSA officials. These are the magical authorities in America and would likely play a role in any story set there.