Pardon my saying so, but our galaxy has like 400 BILLION stars in it, and there's so much there that no Trek show has ever touched. Most of it is Terra Incognita, so why send a mission millions of lightyears when you haven't really even finished exploring your own neighborhood?
yes, you're right. Our galaxy has a lot (maybe 400 billion stars?) and we haven't seen them in the Trek Movie. But why should we?
What we know about the milky way galaxy in Star Trek?
Alpha Quadrant = Xindi? Federation? Cardassian?
Beta Quadrant = Vulcan, Klingon, Romulan, etc
Gamma Quadrant = Dominion
Delta Quadrant = Borg
So let see, there are still plenty stars in there, but that is not relevant for a new series. Why? Because we'll find the same things that we found in the previous Star Trek. It will become the same Star Trek with different Crew and ships.
Plus, we talk about "Trek" here, and "boldly gone where no man has gone before". if we talk about the story of Federation, Galaxy war / politic, then it's fine to go around the Milky Way again. Because Vulcan, Klingon, Romulan, Cardassia, Dominion, Borg, etc are still relevant. But, it is not about "boldly gone where no man has gone before".
But what about if we talking about a new galaxy? Well, it will become a different challenge that can tickle our curiosity. What races we will find there? What civilization? What threat? What kind of Starship that can travel across two galaxies? What kind of crews who willing to leave the safety of the known space we called Milky Way? And what kind of Space that they will find in the other Galaxy?
Everything can be new again. Fresh and empty. The creators can fill it with everything they want; even without reconsidering the previous Star Trek canon.
Plus, the usual warp drive won't do the job. The usual Star Trek technology that we know will not relevant anymore in here. The distant between galaxy, the empty space that they must cross, the new galaxy that 100% different than Milky Way Galaxy..., everything. They give us a great deal of new idea of science fiction (Science Fiction that relevant to us, the 21th century People not the science fiction that has already out of date)
And, by making it different (an expedition of several ships with one is super big / giant ship), we negate the feel of loneliness for the Ship and her crews, without negating the challenge for being far from home.
But well, if you still want the same TNG with different crews and ships, I don't know what to say