It could be that Harry is framed for a serious crime within the wizarding world. Snape, being a respected (although grudgingly) figure in the wizarding community, might be the only one who can prove Harry's innocence. He has the knowledge and the influence, if he chooses to use it, to help Harry clear his name.
Maybe Harry discovers something about his past, perhaps related to his parents, that only Snape can help with. Snape was closely associated with Harry's parents. He might know some secrets or have some information that could help Harry understand a family mystery or some hidden power within him.
One possible reason could be related to a difficult potion she's trying to brew. Snape is a potions master, and Hermione might be facing an ingredient or a process that she can't figure out on her own. For example, a rare potion that requires precise handling of dangerous ingredients.
Snape could be very sarcastic and cold at first. He might taunt Harry about finally coming to him for help. However, as he realizes the gravity of the situation, he could start to take it more seriously and actually use his skills and knowledge to help Harry. For example, he might say something like 'Potter, so you've finally realized that you're in over your head and need my help? Well, don't expect me to hold your hand.' But then he would still start to investigate the problem.
Maybe Snape is involved in some sort of mystery or danger within Hogwarts that has to do with magic spells. Hermione, with her vast knowledge of spells and her in - depth study of the Hogwarts library, could be a great help to him in deciphering old spells or magical texts related to the situation.
One possible reason could be Snape's sense of duty to protect Hogwarts students in general. Even though he dislikes Harry, he might put aside personal feelings to ensure the safety of a student in a dangerous tournament like the Triwizard Tournament. Another reason could be Dumbledore's influence. Dumbledore trusts Snape and may have asked him to keep an eye on Harry during this event.
Perhaps Harry discovers some dark secret about his own past or powers that he thinks only Voldemort can help him understand. For example, if he finds out that his connection to Voldemort is more complex than he thought and that Voldemort holds the key to unlocking or controlling certain aspects of his own magic.
Maybe Snape suspects Harry of being involved in some sort of dangerous activity related to the Dark Arts. Pressing him to the wall could be a way to force Harry to come clean about his actions or any knowledge he might have. It could also be that Snape is acting out of his own insecurities and pent - up emotions towards Harry, which he has been suppressing for a long time.
One possible reason could be his lingering resentment towards James Potter, Harry's father. Snape and James had a long - standing feud during their school days. Since Harry looks a lot like James, Snape might be transferring his old grudges onto Harry.
It could be that Harry demonstrates a great deal of magical talent in a new and unique way in the fanfiction. Snape, being a Potions master and a powerful wizard himself, would be impressed by such talent. Also, if Harry shows an understanding of something that Snape values deeply, like loyalty or sacrifice, it would make Snape have a positive view of him in the fanfiction.
In some fanfictions, Snape could have his own personal vendetta against Dumbledore. Maybe Dumbledore has wronged Snape in the past, and when he sees Harry as a potential victim of Dumbledore's schemes, he decides to help Harry. Another reason could be that Snape has a new understanding of the prophecy and believes that Dumbledore's way of handling it is wrong, so he steps in to help Harry.
One possible reason could be that Snape is starting to see Harry in a different light. Maybe he realizes that Harry has been through a lot and has a moment of empathy. It could also be a sign of a deeper, unspoken connection developing between them in the alternate universe of the fanfiction.