One possible reason could be his feeling of being an outcast. In the original stories, Loki is often treated differently in Asgard due to his nature. He might have felt like he didn't truly belong, which could fuel his hatred. Another reason could be envy. Maybe he saw others in Asgard having things he desired, like power or recognition, and his envy turned into hatred.
He could meet the Frost Giants. Since he has some connection to them already, they might welcome him or at least be intrigued by his presence. He could learn more about his heritage from them and maybe even form an alliance.
It could potentially make his relationships more complex. Some might view him with suspicion, thinking he has ulterior motives for being an Odin fan. Others might try to use this knowledge to their advantage, perhaps to manipulate Loki or gain favor with Odin.
One great 'Asgard loves Loki' fanfic could be 'The Redemption of Loki in Asgard's Eyes'. It shows how the people of Asgard gradually come to understand and love Loki despite his past misdeeds. Another could be 'Asgard's Embrace of Loki' which focuses on the emotional journey of Loki as he feels the love from his home realm.
One possible plot could be Loki leaving Asgard after feeling constantly overshadowed by Thor. He might leave in search of a place where his skills and intelligence are truly valued, perhaps finding a hidden realm where he can start anew and build his own following without the judgment of the Asgardian court.
One reason could be that they see Loki as a victim of circumstances. Maybe they've come to understand that his actions were often influenced by forces beyond his control, like Thanos' manipulations. So they decide to defend him in Asgard.
You can start by searching on popular fanfic platforms like Archive of Our Own (AO3). Just type in 'Asgard loves Loki' in the search bar, and it should bring up a list of relevant fanfics. Another option is Fanfiction.net. They also have a wide range of fan - created stories, and you might find some great 'Asgard loves Loki' ones there.
Loki could enlist the help of some unlikely allies. For example, he might strike a deal with a creature from the realms outside of Asgard. In exchange for a future favor, this ally helps him escape by providing a means of transportation or a way to bypass Asgard's defenses.
One reason could be that Loki feels constantly overshadowed by Thor. In the Asgardian hierarchy, Thor is often seen as the golden child. Loki, being the more complex and misunderstood one, might run away to find a place where he can be his own person without living in Thor's shadow.
Maybe Loki is wrongly accused of a crime in Asgard. The Avengers, having had their own experiences with false accusations, step in to defend him. They believe in justice and know that Loki might not be guilty.