Some examples of TOS showing continuity when they didn't have to...
From "By Any Other Name," referring back to "Where No Man Has Gone Before":
KIRK: What happened to your ship?
ROJAN: There is an energy barrier at the rim of your galaxy.
KIRK: Yes, I know. We've been there.
From "The Trouble With Tribbles," referring back to "Errand of Mercy":
CHEKOV: Under terms of the Organian Peace Treaty, one side or the other must prove it can develop the planet most efficiently.
KIRK: And unfortunately, though the Klingons are brutal and aggressive, they are most efficient.
KOLOTH: I might also add that under terms of the Organian Peace Treaty, you cannot refuse us.
The Organian Peace Treaty is referred to again in "Day of the Dove":
KANG: For three years, the Federation and the Klingon Empire have been at peace. A treaty we have honoured to the letter.
KIRK: We took no action against your ship, Kang.
There's an unexpected reference to Christopher Pike when the computer reads out the record of Mirror-Kirk's command in "Mirror, Mirror":
KIRK: Read out official record of current command.
COMPUTER: Captain James T. Kirk succeeded to command ISS Enterprise through assassination of Captain Christopher Pike. First action, suppression of Gorlan uprising through destruction of rebel home planet. Second action, execution of five thousand colonists on Vega Nine.
From "The Deadly Years," referring back to "The Corbomite Maneuver":
KIRK: Message. From Enterprise to Starfleet Command this sector. Have inadvertently encroached upon Romulan Neutral Zone. Surrounded and under heavy Romulan attack. Escape impossible, shields failing. Will implement destruct order using corbomite device recently installed. Since this will result in the destruction of the Enterprise and all matter in a two hundred thousand kilometre diameter and establish a corresponding dead zone, all Federation ships will avoid this area for the next four solar years. Explosion will take place in one minute. Kirk, commanding Enterprise, out. Mister Sulu, course one eight eight degrees, mark fourteen. Speed, warp factor eight. Stand by.
"I, Mudd" refers back to the end of "Mudd's Women," where Mudd predicted they'd throw away the key if Kirk testified at his trial:
CHEKOV: You know this man, Captain?
KIRK: Oh, do I know him. Harcourt Fenton Mudd, thief...
MUDD: Come now.
KIRK: Swindler and con man...
MUDD: Entrepreneur.
KIRK: Liar and rogue...
MUDD: Did I leave you with that impression?
KIRK: He belongs in jail, which is where I thought I left you, Mudd.
MUDD: And thereby hangs a tale, yes.
KIRK: All right, Harry, explain. How did you get here? We left you in custody after that affair on the Rigel mining planet.
Spock says that his father was an ambassador in "The Naked Time" and sure enough, when we meet Sarek in "Journey to Babel" the next season, he's the Vulcan Ambassador. (JTB didn't contain any references to Amanda being a teacher, though.)
Heck, even in the second pilot, WNMHGB, Kirk tells Gary "I've been worried about you ever since that night on Deneb IV" and in Gary's psych profile, seen onscreen for just a few seconds, his telepathic encounter with the natives of Deneb IV is detailed. And remember that this was done in an age before VCRs and the ability to record or pause programs.
None of these references really
had to be in the episodes, but it's cool that they were. The makers of TOS obviously cared about making the show as consistent as possible.