Other than what's been mentioned, the only dramatic SF show from '60s US prime-time that I can think of was The Invaders, about a man trying to expose the alien invaders hiding among us. It wasn't particularly like Trek, though, aside from having some of the same FX creators.
I'm going to make an odd recommendation: Space: 1999, one of Gerry Anderson's live-action shows. It's kind of a bizarre concept and often quite silly -- the Moon is blown out of Earth orbit and somehow drifts millions of light-years through space and encounters new planets on an almost weekly basis -- but in the first season it almost kinda works in a way, because there was an overall sense that space was a bizarre and incomprehensible realm that challenged our assumptions about reality. It was more a mythic journey in the vein of The Odyssey than a grounded science fiction show, and if you could buy into the trippy, surreal quality, it almost worked. (Maybe it could fall into the "like The Twilight Zone" category as much as the "like Trek" category.) Moreover, there are a couple of things it did even better than TOS: its supporting ensemble cast was better developed, and its technology design and visual effects pretty much left TOS in the dust.
The second season, however, was under new management -- Fred Freiberger, who also produced TOS's third season -- and it was retooled to be more "American" and action-oriented, losing that surreal and philosophical quality in favor of much the same kind of random weirdness you saw in TOS season 3, like aliens happening to have whatever assortment of magic powers was convenient for the plot (teleportation being the msot common by far). It also dumped several of the supporting characters in favor of new ones, including a beautiful alien shapeshifter with a Spock-like intellect. The Freiberger connection does give it a very TOS-like flavor, but mainly like the third season.
Anderson also did an earlier live-action drama called UFO, about a team fighting off an ongoing alien invasion (more overt than the one in The Invaders, but I've never seen it. Unfortunately, Netflix doesn't have the entire series.