A friendship - centered story could start with Hermione stumbling upon Sirius while he's in hiding. Instead of any improper scenarios, they simply start talking. Sirius is impressed by Hermione's vast knowledge and Hermione is intrigued by Sirius' tales of his youth. They become friends who exchange books and stories, with Sirius introducing Hermione to some of the more 'forbidden' magical knowledge that is actually useful in a non - malicious way.