Disclaimer: I own nothing. I am a poor tomato.
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Hermione was relieved she was back in Hogwarts, but not so happy she was with Harry - and Ron.
Although she refused to admit it, she felt smothered by their constant presence, as if she had to be protected. A year had gone by, with the rebuilding of Hogwarts finished by the desperate Shacklebolt, who used funds from Death Eater accounts while the people - no, animals - in question were thereby guaranteed a comfortable life in the new extension of Azkaban.
The Golden Trio had returned to Hogwarts as a sort of bait for Wizarding World, as many parents had begun to home-school their children after fears about Hogwarts.
Harry and Ron didn't need to return; their careers as Aurors were bright, but they decided to come back for 'courses' they didn't need; however, Harry had offered to teach some DADA classes on the side, to the Headmistress' great approval.
But Hermione? She needed to be attached to somewhere; Hogwarts had always been her anchor. After a year of drifting, from gala to matinée, from special appearances to charity balls, she needed a break. And she got one.
She decided to teach something as well, aside from her potions and herbology courses. Something pure and logical; something that didn't remind her of the war. She always lost herself in the world of Arithmancy. And that was what she was going to do.
And somehow, on the way, she was going to fix things with Ron.