She could meet the headmistress of St Trinians. That would be an interesting encounter as Hermione tries to understand the school's rules and the headmistress tries to figure out this strange, book - smart girl. Also, she might meet the troublemakers of the school, those girls who are always getting into scrapes and might see Hermione as either a threat or someone to rope into their schemes.
The plot could involve Hermione trying to reform the St Trinians' school. She sees all the chaos and unruliness there and decides to use her Gryffindor courage and Ravenclaw intelligence to bring some order. Along the way, she has to deal with the unique personalities of the St Trinians girls, who are not used to the strict Hogwarts - like discipline she tries to introduce.
She could meet Bella Swan. They might start off as a bit of an odd couple, with Bella's more reserved nature compared to Hermione's boldness. But they could learn a lot from each other.
In Hermione St Trinians fanfic, Hermione herself is the key figure. But also, there are the St Trinians girls who can be considered main characters as well. These girls might include the girl who is good at sports and is always getting into friendly competitions with Hermione. Another could be the girl who is really into art and tries to get Hermione involved in their creative projects. There's also the girl who is the prankster and often gets Hermione into some unexpected and humorous situations.
Perhaps she'll meet older, more experienced students who can act as mentors. These students could introduce her to secrets of the new school that are not easily accessible to newcomers. They might also help her with any homesickness she has for Hogwarts.
Hermione might also come across some magical creatures that are guarding certain vaults in Gringotts. For example, a dragon that she has to find a way to bypass without getting burned. These encounters would add more excitement and danger to her visit to the wizarding bank.
There's a fanfic where Hermione tries to start a study group at St Trinians. In this story, she faces all sorts of challenges as the St Trinians girls have their own unique way of doing things. It's really interesting to see how she tries to blend in her studious nature with their more rebellious attitudes. And then there's the fanfic that focuses on Hermione using her magic to solve the problems that St Trinians always seems to have, like dealing with rival schools in a very magical and unorthodox way.
There's a chance she meets some old Death Eaters who are lurking around Spinners End. This would put her in a very dangerous situation, but it would also make for an exciting plot. She would have to use her wits and magic skills to outsmart them. Or perhaps she meets some of Snape's old school friends who can provide insights into his character and motives.
One of her friends could be a girl who is also interested in Muggle Studies, which is not really a popular subject in Durmstrang. They meet in the library while looking for books on the topic and start sharing their views on the Muggle world and magic. Also, there might be a young wizard who is against the more aggressive and dark - leaning teachings of Durmstrang and sees Hermione as an ally in changing the school's culture.
Voldemort's remaining Death Eaters could be behind it. They might see Hermione as a threat to their attempts to regain power in the wizarding world.
Madame Maxime, the headmistress of Beauxbatons, would surely play an important role. She might be initially skeptical of Hermione but later come to respect her. Also, there could be a young male wizard at Beauxbatons who is intrigued by Hermione's intelligence and Hogwarts - style magic. He could try to learn from her and in the process, they become friends and face challenges together in the school.