One way Harry might outsmart Hermione in the fanfiction is by being more adaptable. Hermione is used to following rules and procedures she has learned. However, Harry has had to face many unexpected and dangerous situations throughout his life. So, when a new and strange challenge arises in the fanfiction, Harry can quickly adjust and come up with solutions on the fly. He doesn't get stuck in the traditional ways of thinking like Hermione might, and this allows him to outsmart her.
In the fanfiction, Harry could outsmart Hermione by understanding people better. Hermione is often very logical and book - smart, but Harry might be more in tune with the emotions and motives of others. He could use this to his advantage in a situation where they are trying to solve a mystery or deal with a difficult person. He can read between the lines and make decisions based on his gut feeling about people, while Hermione is still analyzing the facts from her books.
Well, he might use his knowledge of magic in unexpected ways. For example, he could use a simple spell in a complex situation that Voldemort wouldn't anticipate. Harry has always been good at thinking on his feet.
Well, in many fanfictions, Harry might use his quick thinking and knowledge of magic. For example, he could use spells in unexpected ways to get around Fudge's plans. Maybe he uses a charm to make Fudge misinterpret a situation or forget an important detail. Harry is known for his resourcefulness, and against Fudge, he could use his skills to expose Fudge's wrongdoings or prevent Fudge from interfering in things that Harry believes are right.
Well, Harry might outsmart Dumbledore by using his unique understanding of the magical world in ways Dumbledore doesn't expect. For example, Harry has a more direct experience with some dark magic items due to his encounters with Voldemort. He could use this knowledge to make decisions that seem counter - intuitive to Dumbledore but are actually more effective in a particular situation.
Well, in many such fanfictions, Harry might use his magic skills. He could cast a powerful stupefy spell on the troll to stop it in its tracks and then quickly lead Hermione to safety. Maybe he also uses his quick thinking and knowledge of the Hogwarts corridors to find a hiding place where the troll can't reach them.
In some fanfiction, Harry Potter might outsmart others by using his quick thinking and knowledge of magic. For example, he could use spells in unexpected ways to get out of tricky situations.
Well, in many fanfics, Harry might use his quick thinking and swimming skills. He could have found Hermione using his magical connection to her or by following the signs in the underwater maze. Maybe he used a spell to break whatever was binding Hermione and then swam her back to the surface.
One interesting strategy could be his use of misdirection. He might make others think he's going one way while actually doing something completely different. This keeps his opponents guessing.
In the fanfiction, there could be a situation where they are dealing with a magical creature. Hermione starts to research all about the creature's habits and weaknesses from her books. But Harry, having had some prior experiences with similar creatures, uses his practical knowledge. He approaches the creature in a different way than Hermione would have expected based on her research, and is able to tame or defeat it more easily, thus outsmarting her.
A common plot device is misdirection. Harry might create a false lead or distraction that makes Snape focus on the wrong thing. For example, he could stage a minor 'accident' in the corridor that draws Snape's attention while he slips away to do something else he's not supposed to. Another device is using his friends' unique abilities. If Hermione has found a loophole in a rule, Harry can use that knowledge to outmaneuver Snape who is always strict about the rules.
She's Hermione, so she's super - organized. She probably makes a strict schedule. She studies when Harry is asleep. And she uses magic to her advantage, like a spell to keep an eye on Harry while she's poring over her books.