after the events of First Contact
I'd love that time period, but would still like a starship focus. We would see and hear on occasion about what's happening on Earth, but not all the time.
Jettison the whole ST Enterprise premise (alternate universe) and have starships out exploring and colony ships departing right after Cochrane proves his engine works. This would place the series only about six decades from today.
The starship would be looking for colony world for Humans who wanted to leave Earths war torn regions behind, and looking for resources and new technology to help Earth rebuild, and for new trading partners and allies.
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Different idea would be (my vision) a post-Dominion War series. Many of the Federation members are less than impressed with the Federation Council's leadership and conduit during the war and the Federation is beginning to shed members. They're are going ...
1) independant.
2) banding together with other former Members.
3) Joining with other interstellar alliances (who were always there).
The former area that was the Federation is now much more dynamic, weaker in some way, stronger in others. Where there was once one Federation, now there are more than twenty.
This version of Star Trek would be more political and more interspective.
We'll find out that many in the old Federation were unhappy with the way things were run, the iron cast rules created before most of the Membership originally joined, the power block of the founding six Members (and their unique ability to veto any decision), and the prime directive and the way it evolved between the 23rd and 24th centuries.
And especially the way Humans dominated Starfleet.