When TMP came out, the actors had aged ten full years, and yet TMP seemed to suggest it was under three years since the end of TOS.
...At a minimum. At least 2.5 years after TOS, but quite possibly a decade or more after the show. Indeed, the movie suggested that a Voyager probe would be "over" 300 years old at that point, which would only be true if the movie took place 300 years after the premiere date or later (first two Voyagers launched in 1977, movie written with the intent of coming out in 1979, movie events supposedly taking place in 2279 or perhaps a bit later).
ST:TMP seems to establish that there was no second 5ym for Kirk between TOS and the movie - otherwise, Kirk would have said his CV featured two such missions, or "ten years out there dealing with unknowns like this", or something to that effect. If the movie is intended to take place in 2279 (or before 2278, to satisfy some fine-tuning introduced in TNG "Cause and Effect") and the sequel in the mid-2280s ("a while" after the 2283 date on McCoy's ale bottle), then there's room for a second 5ym there, but just barely. And dramatically one wouldn't expect such a mission there, because Kirk in the second movie is desk-bound and bored out of his skull, if possible even more so than in the first movie.
Perhaps there was a 2nd 5ym without Kirk? Perhaps with Spock in command of the hero ship, perhaps with some other character altogether? This could have taken place before, after or basically even during TMP. All we know is that Kirk is highly unlikely to have been involved in it - unless we assume TMP took place earlier than the events themselves would indicate.
There's always the slot after ST2-5 but before ST6: supposedly between the mid-2280s and 2293. Kirk would be a lowly Captain again, everybody would assuredly have the cool ST2 uniforms rather than the TMP pajamas, and the plotlines could involve things introduced in ST2-5, such as Carol Marcus or Saavik or renewed Klingon aggression and/or detente, or all sorts of hilarity with the reborn Spock.
Timo Saloniemi