One possibility is that he wakes up and discovers new allies. The story could then revolve around him building relationships with these new characters while still holding onto his old values. He may also have to deal with the fact that the world has moved on without him and find his new place in it.
Maybe it's an old enemy who has somehow survived and is now more powerful. They could be seeking revenge on Captain America for past defeats. For example, Red Skull could have found a new way to return and cause trouble.
Well, in some fanfics, they might make Captain America seem like a jerk to create an interesting plot twist. Maybe they show him being overly self - righteous or making bad decisions that go against his usual heroic image.
Well, perhaps Captain America could be on a mission to seek an ancient Asgardian weapon to help in the fight against a new threat on Earth. He might meet Thor and the other Asgardians, and have to prove himself worthy in their eyes, just like he did in his own world.
He could also meet Odin. Odin might view Captain America as an interesting mortal from Midgard and might have some words of wisdom or a task for him to complete during his stay in Asgard.
The Night King could be a great antagonist. His army of the dead and his own powerful magic would pose a huge threat to Captain America. Cap's used to fighting human foes, but the Night King is something entirely different.
There's also 'Captain America: Reborn'. In this fanfic, Steve Rogers has to face a new threat that is specifically targeting his ideals. It's a great read as it explores his unwavering beliefs and how he inspires those around him even in the face of seemingly insurmountable odds. It also has some unique interpretations of his powers and abilities.
Captain America could fit in by sharing his sense of morality and leadership. In the TOS universe, which often deals with complex moral and ethical dilemmas, his straightforward and noble values could provide a different perspective. He might be involved in diplomatic missions, using his charisma and integrity to negotiate with alien races. Also, his military experience could be applied to strategic planning during the crew's missions.
Well, perhaps a plot where they accidentally end up in an alternate dimension. In this dimension, their powers are different. Captain America might have some super - speed and Superman might have a more tactical mind like Cap. They have to figure out how to get back to their own dimension while also dealing with the villains of this new world. They learn from each other's skills and grow closer as friends throughout this adventure.
One possible plot could be that Captain America is in a situation where he accidentally injures a fan during a chaotic battle. Maybe he's fighting a powerful enemy in a crowded area and his shield goes astray. He then feels extremely guilty and tries to make amends by using his resources to help the injured fan recover. He might also face public backlash for the incident which adds to his internal struggle.
One idea could be exploring how the fractured Avengers try to regroup on their own. For example, Captain America could be leading a small, underground team that focuses on helping those affected by the Sokovia Accords in a more discreet way. Meanwhile, Iron Man might be working with the government to create a new, more controlled version of the Avengers.