Is six months really long enough for Marcus to have recruited Khan and sent him on a couple of missions?
Actually, I think it's mentioned by Khan that the Botany Bay was found when Starfleet did a sweep of nearby sectors for other threats some time after the Kelvin was destroyed, so he was in Section 31 for longer than a year.
But as a result of the destruction of Vulcan, your Starfleet began to aggressively search distant quadrants of space.
My ship was found adrift. I alone was revived.
I looked up "John Harrison." Until a year ago, he didn't exist.
John Harrison was a fiction created the moment I was awoken by your Admiral Marcus to help him advance his cause.