If you don't mind an off-topic question: Why did Kirk become an Admiral directly after Captain, and not a Commodore?
Because the commodore rank was apparently phased out around that time. In real life, the US Navy stopped using the commodore rank in 1947, replacing it with rear admiral (although since the '80s commodore has been used as a title
for senior captains in certain positions), and apparently the makers of TMP and later Trek productions decided to follow that precedent, perhaps feeling the commodore rank was too old-fashioned. TMP did include a reference to "Commodore Probert" in background dialogue recorded for the Epsilon 9 sequence, but that was just filler dialogue that wasn't meant to be clearly audible. There are also screen graphics listing commodores in TNG: "Conspiracy," but again those probably weren't meant to be clearly legible, and aside from that there are no references to Starfleet commodores at any point after TMP.