Even the first season had moments of levity within some of the most serious episodes; this isn't strictly a matter of changes between seasons 2 and 3. For example: Spock's various statements of what the odds against them were (and Kirk's reactions to them) in "Errand of Mercy"; the Chinese rice-picker story in "City on the Edge of Forever." And the less heavy episodes were full of moments that could raise a smile: "I'm going to have a son" in "Tomorrow is Yesterday"; "I'm about to start
devoting my whole life to it" in "Mudd's Women"; etc. (I'm leaving out "Shore Leave" and "Squire of Gothos" as they were often overtly comic.) There is, as far as I can remember, no such moment in any season 3 episode, assuming again that "Spock's Brain" is unintentionally funny.