Also, keep in mind that someone referring to him as "Commander Spock" does not preclude the possibility of him actually being a Lieutenant Commander. It seems fairly common for the Lieutenant Commander rank to be shortened to just Commander in conversation in Starfleet, and I believe I remember hearing that was true of the modern U.S. Navy as well. For another example of this, Picard and others frequently referred to Data as "Commander Data" even though his rank was Lieutenant Commander.
Hence, we need to rely on "Let That Be..." as the definitive indication that Spock was ultimately promoted to Commander.
Back to the "second officer" thing...
I looked up the transcription of "The Enemy Within", and, sure enough, in the Bridge scene where Spock makes a V.O. log entry, he does indeed identify himself as "Second Officer Spock":
TAKEN FROM "THE ENEMY WITHIN":
Captain's Log, stardate 1673.1. Entry made by Second Officer Spock. Captain Kirk retains command of this vessel, but his force of will rapidly fading. Condition of landing party critical. Transporter unit still under repair.
Sooooooo... at that point Spock was regarding himself as the "second officer", not the first. This leaves a little wiggle room for someone else (unseen) to be serving as the Enterprise's XO. Also note: during Year 1, Kirk repeatedly refers to Spock as "Science Officer", obviously Spock's primary title.
I do not believe Spock is called "first officer" until he makes a log entry in Sickbay (prior to mind-melding with Dr. Van Gelder) in "Dagger of the Mind".
I always thought this was strange, but there could be an explanation. You'll recall that Spock's command of the Galileo in "The Galileo Seven" was regarded as his "first command". It could be that Spock was in training for the XO position and this deep space expedition was a necessary assignment as part of that preparation for promotion. It is not until a few episodes later that Spock is referred to as both "first officer" and "science officer":
TAKEN FROM "COURT MARTIAL":
COMPUTER: Spock, serial number S179-276SP. Service rank, Lieutenant Commander. Position, First officer, science officer. Current assignment, USS Enterprise. Commendations, Vulcanian Scientific Legion of Honour. Awards of valour. Twice decorated by Starfleet command.
Unless I missed something, this is the first time Spock is officially referred to as "first officer" by someone/something else; It could be that Kirk gave him the provisional title prior to "Dagger..." pending the Fleet HQ nod on his promotion sometime later.