Seems to me he did, especially if he bought the Coke
beforehand in order to use it as a prop all the way, if necessary.
Carrying the already-opened Coke, get to the Second Floor Lunch Room and hang out there,
expecting to be encountered ("No, I didn't hear any shots," or "I work here"), when they leave, walk through Mrs. Reid's office to its front door and exit through it, walk down the wooden steps to the front floor, maybe hang out in the "lobby" for a few seconds, then walk to the rear of the building and exit it through the loading dock door.
Instead of looking suspicious by immediately walking or running away from the building, walk along the side of the building towards the front (as alleged by B. W. F.), cross over to the other side of the street, ... and, well, "take it from there".
-- MWT