Using the right key words I suspect a search engine will retrieve some intriquing examples. Just be aware a percentage of them will be "rule 34" fantasies.
Some of the better examples (in my admittedly biased opinion) came from the late Paul S. Gibbs. He created a site in 1997 titled simply "Two Star Trek Novels". Gibbs wrote a 500 page manuscript he tried to submit to Pocket Books. Needless to say, it was rejected for several reasons. It didn't focus upon any of the leading characters; it used material from the animated series, namely, M'Ress, which was in a state of legal flux due to Filmation being bought by a European holding company, and conditions were not reverted to the status quo by the end of the story. Rather than hide the material within a desk drawer, he opted to post his material and see if anybody might be curious.
In brief, the first story depicted M'Ress growing frustrated aboard the Enterprise, suspecting her position in communications might not let her advance in rank. So with Kirk's reluctant blessing, she transfers to a smaller, older ship as part of the science division using her studies in archeology. Alas, she discovers she she has landed in a nest of human supremicists and her tour of duty becomes a living hell.
That narrative and its follow-up no longer exist on-line in their original form. Gibbsbecame almost obsessively frustrated over Pocket Books' rejection, not for quality, but for not adhering to (what he considered arbitrary, limiting) guidelines. I suggested if it bothered him that much, just drop the Trek references, create a new backstory and try submitting it to other publishers. He was reluctant at first, saying he was more comfortable "playing in someone else's sandbox." Finally, in desperation, I sent him a drawing of felinoid in a jumpsuit very different from stuff we normally see in Trek, wearing an ocular positioned readout device and scanning apparatus worn as a glove. The next day, he replied, "OK, so, what should they use instead of 'warp drive'?"
Thus he started a several month project where he created a different galactic community that used naturally occurring rifts in space/time to "displace" their ships from system to system. Her captain was an older Lloyd Bridges type personality. Her best friends hailed from India and a low gravity planet, and the chief engineer looked like an elephant with a head like Cthulhu. M'Ress became Ehm'Ayla and her species became Sah'aarans.
And though we lost Gibbs some ten years ago to pancreantitus (because he felt he didn't have the money to afford a doctor), his material lives on at the...
While it's no longer a Trek story, 90+ percent of that first story in terms of motivations and goals is based upon the older manuscript. His second story required more reworking to excise the Trek elements and the material that followed was all his own, owing nothing to Roddenberry's creation.