I rewatched that scene too, just to revell in the closeup - and noticed something else!
Just prior to that scene, Spock meets Kirk coming out of a (implied) turbolift and hands him a phaser before they proceed to Environmental Engineering.
What's curious is that it's the same corridor setup (red doors to the left, blue airlock doors at the end) as was featured in LTBYLB a couple of weeks later, with only a couple of minor adjustments.
I say "implied" turbolift because while the doors are the correct colour, the carpet is not, plus the inner doors are missing. In fact, this is just the side door to the Briefing Room with a change in door colour. but who is going to notice those differences?
This side door was used rarely throughout the show because it requred reconfiguring the wall panels on the Briefing Room Wall but at the time of WOAE, that reconfiguration was still in place from
Day Of The Dove so tweaking it for WOAE would have been relatively cheap & painless.
I've always thought that the LTBYLB was unique, engineered to best suit the filming needs of a chase through the corridors. However, since both episodes were directed by Jud Taylor it seems he simply capitalised on a useful configuration which was left over from his previous episode! Or was he just a great forward planner?
Either way, a fascinating glimpse into the production!
