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.
One possible plot could be that Jon Snow uses his military skills learned from the Night's Watch and his newfound alliances. He might start by uniting the North completely, including the wildlings. Then, he could gradually move south, facing challenges from different Houses. His claim to the Iron Throne could be based on his Targaryen heritage, which he may reveal at a crucial moment to gain more support.
The White Walkers can still be the main antagonist even if he leaves the Night's Watch. After all, they are a huge threat to all of Westeros. Jon might think he can fight them better outside the Night's Watch, but they are constantly advancing and could target him specifically for leaving his post. Maybe they see him as a key figure in the coming war and want to eliminate him before he can gather enough strength to oppose them.
In some fanfictions, he might leave due to a love interest. He could meet someone who makes him question his vows to the Night's Watch. This person could be from a different region or have a special connection to him. For example, she might be a noblewoman with a hidden agenda, but Jon is drawn to her and decides to leave the Night's Watch to be with her and get involved in the political intrigue of the Seven Kingdoms.
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.