Ah that "ladder access" corridor scene, quite a problem indeed! While the implicatation is that they've just climbed down it from the EMM, as you've found that obviously couldn't be the case!
Regarding the EMM ladder providing access to the Engine Room however, the episodes tend not to support this - in I, Mudd
Norman goes down the ladder and round through the main doors. In Is There In Truth
Scotty follows the same route. In both cases the characters would have taken a more direct route, if they could (which suggests they couldn't). Perhaps that side of the Engine Room is blocked with equipment?
An alternative solution is to have the EMM ladder keep on descending to the deck below
the Engine Room - that way, it really can
connect to the corridor!