In terms of his romantic relationships, if there was a budding relationship with Hinata, for instance, his new status as a jounin could either strengthen their bond as she admires his achievement, or it could create some distance as he has more duties and less time for personal matters.
In this fanfic, Naruto could finally gain the respect he truly deserves as a Jounin. He might lead his own team and pass on the lessons he learned from Kakashi and others. His unique jutsu like the Rasengan could be further developed and he could use it in more creative ways during missions.
Naruto's relationship with the younger generation would be very different. As a Jounin, he would be a role model. The kids in the village would look up to him even more. He could inspire them to work hard. His relationship with Hinata might also evolve. Maybe he would finally have more time to focus on their relationship now that he has achieved a higher rank and more stability in his ninja career.
Perhaps Naruto, as a young Jounin sensei, has to deal with a student who has a dark past similar to his own. He has to figure out how to help the student open up and trust him. Naruto uses his own story of growing up alone and being shunned by the village at first to connect with the student. He shows the student that they can rise above their past and become a great ninja.
Perhaps the plot could revolve around Naruto's internal struggle. Even though he has achieved the Jounin rank, he still doubts his own abilities. He could face challenges from other Jounin who don't believe he's worthy of the rank. Meanwhile, he discovers a new and dangerous enemy that is targeting Konoha's Jounin. Naruto has to overcome his self - doubt and use his skills, like his Rasengan and Sage Mode, to defeat this new threat and prove his worth as a Jounin.
Kakashi could still be an important ally even though Naruto has become a Jounin. Kakashi has a wealth of experience and knowledge that Naruto could still learn from. He could offer advice and assistance when Naruto faces tough challenges in his new role.
Well, Asuma is also a character that shows up a lot in these fanfics. His relationship with his students and his smoking habit give fanfic writers a lot to work with. And then there's Kurenai. Her genjutsu skills and her role as a female jounin can add unique elements to the stories. There are also some original characters created by fanfic authors that sometimes become popular within the 'naruto jounin fanfic' community.
In this kind of fanfic, one interesting storyline could be Naruto facing new and more powerful enemies as a Jounin. He might have to use his unique skills in more strategic ways. For example, his shadow clone jutsu could be used to deceive opponents on a much larger scale.
One interesting plot could be a jounin who has a secret mission to infiltrate an enemy village. He has to use his unique jutsu and skills to avoid detection. Another plot might involve a jounin training a group of genin in a very unorthodox way that challenges the traditional shinobi training methods.
One common theme is self - discovery. Since backup jounins are often in the background, these fanfics usually explore how they find their own strength and purpose. Another theme is loyalty. They have to show loyalty to their village and comrades even when they are not in the spotlight.