You could get a soft reboot in the early 25th century. For one thing, galactic politics might have become relatively quiet, because the major powers have all been traumatized. So for awhile a peace of exhaustion might prevail. Within the Federation, it is a period of Reconstruction rather than a period of Utopia. Smugness is now in short supply. Hopefully, the same is not true of optimism (a key Trek trait). There has been enough time for stylistic changes, such as clothing styles, or hair styles. Conceivably, there could be frontier areas on the Federation's periphery where new colonies are being founded. With a bit of exploration beyond by Star Fleet. However, new technologies may still be in an experimental stage, not yet ready to transform society.
I see the word reconstruction coming up a lot, but if they picked up TNG-era Trek and accounted for the passage of time in the real world, would reconstruction from the Dominion War still be a current thing? It's been 19 years and counting since it ended. There'd "currently" be freshmen in the Academy who hadn't been alive during the war.
But you needn't set the series that many years after the war; the series could start somewhat earlier.
There have been a few series which show teams investigating strange phenomena: The X-Files, Torchwood, Project UFO.... It turned out that there was considerable story potential for episodes set on Earth.
And during Reconstruction, there was a frontier in the West. So during a similar period, we can imagine a frontier with something of a Wild West atmosphere, while Star Fleet is exploring just beyond. And Star Trek being Star Trek, we can imagine a few planets populated by transplanted humans-deposited there by aliens such as the Preservers-leading to subgenera such as planetary romance; perhaps a lost colony; artifacts left behind by super civilizations; conflict with contemporary aliens; trans dimensional beings; etc......
People have commented that some Trek episodes and novels were similar to The Twilight Zone. So there was the idea that a small team would go out in a ship to investigate strange phenomena. Which might often mean traveling around a sprawling, diverse frontier (see my earlier post). Mulder and Scully, meet Star Trek.
I really don't like the fractured Federation idea. It's what 90% the proposals for Trek TV shows over the years are as well. What I like about Trek is that it is optimistic and hopeful and shows a better future. And in that, why not explore one of the single most unexplored elements of Star Trek... the Federation? We have NEVER seen a Federation city actually in proper action outside of the Kelvin films. Yorktown really is the closest we've seen to what life is actually like for civilians in the Federation. How about a Star Trek show that doesn't actually focus on Starfleet but instead focuses on Civilian life. Instead of following Starfleet, we follow maybe some archaeologists or The Daystrom Research Institute? You can still have exploration and exploring new worlds, but here is an idea, you have a Daystrom Sociologist who studies the cultures of Federation aliens, and the series is about this scientist, visiting different Federation worlds and aliens species and living among them for a time, learning about their societies? You can use that premise to basically spend 2-3 episodes on each planet with new characters, having those characters go through some story or arc, then our character leaves to another planet and we get another story. Something like Gulliver's Travels or Kino's Journey. This reminds me of once talking to a Sci-fi Writer who said one of the biggest missed potentials of Star Trek, is say take the Enterprise-D, if used to it's full potential (Crew in the thousands and thousands), you've basically got a flying CITY in space. Why doesn't Star Trek actually do anything with that concept?
I agree with one comment by Donker-an optimistic streak is at the core of Star Trek. However, we did get a glimpse of a Federation city in "Prelude to Axanar". The video is online.