In a fanfic, Jon Snow could be called a bastard during a tournament. Rivals from other houses might use it to try and get under his skin before a joust or a melee. Also, in a tense council meeting where his status and right to be part of important decisions were being questioned, the term 'bastard' could be thrown at him. Another instance might be when he's on a journey and meets some unfriendly travelers who learn of his identity and call him 'bastard' to assert their supposed superiority.
In a fanfic set during his training days, an envious trainee could call him 'bastard' to try and disrupt his focus. Another example could be when he's in a negotiation with a rival group, and they use it as a bargaining chip to try and get more concessions from him.
Jon Snow could potentially have allies like the Mormonts. They have always been loyal to the Starks. Maybe even some of the Dornish houses might side with him. They have their own reasons to oppose the current power structure in Westeros. And of course, his friends from the Night's Watch who left with him or believe in his cause could be his allies in this conquest. He might also gain the support of some of the maesters who see the value in his leadership for the future of Westeros.
Daenerys Targaryen could be one to help. Since she has experience with dragons, she could give Jon Snow advice on how to care for the egg and hatch it.
It could be a young and untainted Frey. Maybe a niece or a cousin of the more infamous Freys. One who has not been involved in the past atrocities committed by her family and has a more noble heart.
Daenerys Targaryen is often paired with Jon Snow in such fanfictions. Their combined power and the potential for a different kind of relationship makes it a popular choice among writers. Another character is Arya Stark. Their sibling bond can be explored in new ways when Jon is reincarnated, like Arya helping Jon with her assassin skills in a different plot line.
The North would surely support him. They have always seen him as one of their own and a great leader. Also, the wildlings might back him, considering he has a history of working with them and understanding their plight. Davos Seaworth would most likely be on his side as well, given his loyalty to Jon in the past.
Daenerys Targaryen is often paired with Jon Snow in these fanfictions. Their connection in the original series, along with their both being strong leaders, makes for an interesting dynamic. They could rule Westeros together in a new era of peace and prosperity after Jon's rebirth.
The wildlings might be his allies. After all, Jon Snow has had interactions with them before. They are fierce fighters and could fight alongside him against a common enemy.