This is the first time I've heard this suggested, is there anything to back up Spock being the second officer?
Thee are scenes in the original pilot film "The Cage" that didn't make "the cut" into "The Menagerie". After Number One and Colt find themselves in the cage with Pike and Vina and bicker, the scene in "The Cage" cuts back to the bridge:
[Bridge]
SPOCK: Address intercraft.
GARISON: Open, sir.
SPOCK: This is the acting captain speaking. We have no choice now but to consider the safety of this vessel and the remainder of the crew. We're leaving. All decks prepare for hyperdrive. Time warp factor.
TYLER: Mister Spock, the ship's controls have gone dead.
(The lights go out)
SPOCK: Engine room!
GARISON: Open.
SPOCK: Mister Spock here. Switch to rockets. We're blasting out.
PITCAIRN [OC]: All systems are out, bridge. We've got nothing.
TYLER: There's nothing. Every system aboard is fading out.
And in a later scene:
Bridge]
(The senior officers have taken the navigation console apart)
SPOCK: Nothing. But for the batteries we'd lose gravitation and oxygen.
TYLER: The computers!
(The monitor shows a montage of images - space capsules, the Moon, maps of Earth)
TYLER: I can't shut it off. It's running through our library. Tapes, micro-records, everything. It doesn't make sense.
SPOCK: Could be we've waited too long. It's collecting all the information stored in this fly. They've decided to swat us.
http://www.chakoteya.net/StarTrek/1.htm
Clearly Spock is the senior officer in command, and he even calls himself the "acting captain". Thus he must be a more senior lieutenant than Jose Tyler and any other lieutenant aboard as a line officer. Dr. Boyce probably doesn't have any command rights and thus could have a higher rank than Spock.
A small indication that Spock is in command after Pike and Number One are captured survives in "the Menageire":
[Transporter room]
PITCAIRN: Sir, it just came on. We can't shut the power off.
SPOCK: Mister Spock here.
TYLER [OC]: All power has come on, Mister Spock. The helm is answering to control.
(first Colt, then Number One are beamed aboard)
??: The captain.
http://www.chakoteya.net/StarTrek/16b.htm
So certainly anyone who accepts "The Cage" as canon, and it seems to be included in official canon, has to accept that somehow Spock was already third in command by then, probably just four years after first being commissioned an ensign.