Over the years I've come to think of WNMHGB happening not long (some months) after Kirk takes command of the Enterprise from Pike. I imagine this happening at some far off starbase rather than back on Earth. And hence WNMHGB could happen about 1-2 years before the events of first season TOS. I base this not only on the appearance of the ship and the uniforms, but also on the fact there is no opening narration in the episode's opening credits unlike the rest of the series which refers specifically to the 5-year mission.
I'm figuring about a year and a half before the series proper to account for sufficient time for a few more events including a ship's refit of some months leading to what we see at the behinning of the first season.
I'm figuring about a year and a half before the series proper to account for sufficient time for a few more events including a ship's refit of some months leading to what we see at the behinning of the first season.