^Since Jeremy Renner was originally intended to be the new lead, he may well have been signed to do more than one movie.
Of course, a sense of connection between installments wasn't really a feature of the original show. The continuity, typically of '60s TV, was essentially nonexistent. The team members could show their faces on national or global TV at the climax of one episode and yet still be able to function as secret agents the next week. The team composition changed frequently; originally the intent was that a different team of specialists would be chosen every week, and although eventually the main cast members ended up being used much more regularly, cast changes between seasons happened frequently and without a word of explanation (until the final season, which also saw slight traces of continuity introduced for the first time).