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.
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.
One common theme is redemption. Loki is often seen as a character who has made mistakes, and in these fanfictions, Asgard's protection of him is part of his journey towards redemption. For example, Thor's unwavering support helps Loki to face his past actions and strive to be better.
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.
He might first find a place to hide, like an abandoned building. There, he starts to realize how difficult it is to be on his own. He has to find food and water by himself.
In the fanfiction, Loki's hatred for Asgard might make him more of an anti - hero. In the original canon, he has complex relationships with Asgard, but not always pure hatred. Here, his actions are likely to be more extreme as he is driven by this strong negative emotion.
I also like 'Asgard's Love for Loki'. In this fanfiction, Odin, despite his past mistakes regarding Loki, decides to be more protective. He uses his power and influence to shield Loki from external threats and also tries to mend their broken relationship. It's an interesting take on how Asgard can step up to protect one of its own who has often been seen as an outcast.