My head canon agrees that Mitchell was already on the Enterprise serving with Spock for 2-3 years before Kirk became Captain on the Enterprise. Both are Lt. Cmd.'s serving under Pike, so, who would Pike want to be first officer? To me, Spock is Pike's obvious choice due to his long, 11+ year relationship. I also recognize that Spock does not seek command, so, maybe Pike takes Mitchell on because of this, but I see this as a weaker argument. Maybe Starfleet politics got involved where Mitchell was their choice as first officer over Spock because "reasons".(BTW, my personal headcanon is that Mitchell was actually assigned to the Enterprise before Kirk took command. That's how he knew Spock "for years" by the time of WNMHGB. The first command that Kirk requested Mitchell for was the cruiser he commanded before the Enterprise, but Mitchell was unavailable for that position. But as luck would have it, Kirk took command of the Enterprise himself a couple of years later. So at the time of WNMHGB, Mitchell actually knew Mr. Spock at bit better than Kirk did at that point.)

Years prior to that, Kirk and Mitchell are serving together where Mitchell takes a poisoned dart meant for Kirk from a rodent thing on Dimorus and saves Kirk’s life. In my head canon, I put this event during Kirk's first command where he requested, and got, Mitchell to serve with him. What duty position did he have? Helmsman? First Officer? At this point I use the actors age difference (which I know is weak in TV casting): Bill Shatner is 6 years older than Gary Lockwood, but plays Kirk as two years younger than his real age (The Deadly Years), so I conclude a 4 year gap minimum. This 4(+) age gap is confirmed by Mitchell, "Hey man, I remember you back at the Academy. A stack of books with legs. The first thing I ever heard from an upperclassman was, watch out for Lieutenant Kirk. In his class, you either think or sink." I assume Mitchell is an underclassman and not in the faculty. If Kirk is now a young Commander on his first command, then Mitchell is probably a Lieutenant. To have Mitchell a first officer at this point seems unlikely since it would exceed the career path of even Kirk, so, not first officer is my conclusion.
Shortly later Kirk and Mitchell are separated where Mitchell ends up on the Enterprise. Why? Does Una Chin-Riley (aka Number One) leave the Enterprise (circa 2261 to 2263) and Mitchell is brought in to replace her as the new helmsman? In any case, the helmsman job on the Enterprise is a plumb assignment over serving on a smaller vessel, hence a good reason Mitchell transferred. This transfer most likely came with a new promotion to Lieutenant Commander, too (thanks to Kirk). Bringing him in as both helmsman and first officer seems a big stretch to me. YMMV

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