Interestingly I could almost argue that this Arc’s Kishotenketsu structure correlates to its episodes that aired in Gregorian Calendar years 2001-2004.
Of course 2000 can be argued to already be part of the Vermouth era. The character of Vermouth is technically introduced in episodes 176-178 Reunion with The Black Organization in January of 2000. And then 188-193 The Desperate Rival can be considered connected to it. But that's all stage setting, you could follow the storyline of the Vermouth Arc proper without really having seen them. Heck a 1999 episode is arguably more relevant to the plot 170-171 The Blind Spot in the Darkness.
I will be mentioning some episodes I like from this era that aren't part of the Arc, but I still haven’t seen every episode even in this area so my knowledge is incomplete.
2001 contains the Ki of the arc.
226-227 The Battle Game Trap is perhaps still just stage setting like the 2000 episodes, but the character it introduces is very important.
Not relevant to the arc but 228-229 The Murderous Pottery Class is one of the show’s best Columbo episodes.
2001’s film was the first and for a while only movie featuring the Black Organization, Countdown to Heaven. Neither Vermouth or any other distinctly Vermouth arc characters are in it, but it’s also the best Haibara movie and does a good job at setting up where she is as a character when we enter this saga.
230-231 The Mysterious Passenger is the episode that properly initiates the storyline, it establishes a key aspect of how Vermouth operates we didn’t know about before so it can then introduce three suspects for who among those the main cast interacts with she could be. It’s also one of only three Cases from this arc included in Netflix’s Black Organization Collection and thus with an official Dub.
253-254 Metropolitan Police Love Story 4 is the only chapter in that ongoing storyline also tied into the arc since two of the suspects appear in it.
258-259 The Man from Chicago introduces another relevant character. Putting all the pieces finally on the board.
The 2002 episodes are the Sho, further developing what the Ki introduced.
271-271 The Secret Rushed Omission.
275-275 Truth of the Haunted House features characters but is perhaps not too relevant.
277-278 English Teacher vs Great Western Detective.
279-280 Hooligan’s Labyrinth is an often overlooked important episode to Hiabara’s character development.
284-285 Chinatown Deja Vu in the Rain sets the stage for the following episode.
286-287 Shinichi Kudo’s New York Murder Case is the second case of this Arc included on Netflix.
That’s all the key Vermouth Arc episodes of 2002. 301-302 near the end of the year is a neat informal continuation of the Sato and Takagi storyline. Episode 303 is a TV Original but I like it as a stand out Mitsuhiko episode.
The movie for 2002 was The Phantom of Baker Street which I have mixed feelings on. I ultimately rank it among the lowest but there is stuff I like in its first half. It’s interesting in how it was the last film to be Cel Animated and is an early example of a trapped in VR Video Game from the same year as .hack and the original SAO web novel.
2002’s OVA was the second one, 16 Suspects. Which is a neat relic of how Shiratori was characterized early on that I think has maybe been forgotten.
I began writing this just as an argument for 2003 being the best year of the Detective Conan Anime. The year began with episode 304 The Trembling Police Headquarters: The 12 Million Hostages, which was a special two hour long episode. During this era Detective Conan began most years with a 2 hour special so that isn’t unique and this isn’t the best of them, but it’s also far from the least which is impressive since none are bad.
Remember those rants I did on how much of the Vermouth Arc was skipped by the Netflix collection? Well it turns out a lot of the most important stuff that should have been included aired in 2003. The important plot developments here are the Ten.
To list them again those episodes are.
307-308 On The Trail of a Silent Witness.
307-311 Contact with The Black Organization.
329-330 A Friendship can’t be Bought.
335-336 Secret of the Tohto Film Development Studio
338-339 The Four Porches.
340-341 Hidden Bathroom Secret.
343-344 The Convenience Store Trap. Which were also the last episodes to air in 2003.
So basically Netflix skipped an entire Act of the storyline.
I haven’t even seen every episode from 2003. It looks like there were some Heiji cases, and an Eri appearance and some other fun filler episodes.
331-332 The Suspicious Spicy Curry is on my probably still incomplete list of Columbo episodes.
The movie for 2003 was the 7th Crossroad of The Ancient Capital. It was the first to be Digitally Animated and the last directed by Kenji Kodama who did all of them before this. It’s the first movie where Heiji and Kazuha are co-leads for the entire film. It’s pretty good.
The Annual OVA for 2003 was also a Heiji and Kazuha one.
And then the two and a half hour finale of the arc that opened 2004 is the Ketsu. Episode 345 Head-to-Head Match with the Black Organization: Dual Mystery on a Full Moon Night. This episode is dubbed on Netflix, and it’s Dub is great, you should watch everything that’s on netflix.
The next episode 346-347 Find The Buttock's Mark served as a bit of an epilogue to the arc in how it opens at least.
361-362 Teitan High School’s Ghost Story also serves as a bit of an epilogue.
2004’s late Winter-Early Spring had a lot of Kaitou Kid material.
The 4th OVA in March was Conan, Kid and the Crystal Mother which adapts a chapter of the Magic Kaitou Manga but writes Conan into it as he wasn’t there originally.
A TV Special to promote the next few examples titled Time Travel of the Silver Sky. It’s kind of a Kid focused Recap episode with a framing device starring Agasa.
His fourth appearance in the TV Anime, but arguably second proper Kid episode, 356 Kaitou Kid’s Miraculous Midair Walk.
And his second movie, Magician of the Silver Sky, which I rank as an A tier film.
There is also more epiloguing of the Vermouth Arc in Kid’s 2005 appearance 394-396 Big Adventure in The Electric Mansion.
Which is in turn followed up on in 398-399 The Strange Family’s Request.