We're yet another generation since we both recognize him from TOS. Most of us who would look at this as our "Elisha Cook" memory are, at least, likely to have seen one of his other roles when one of his films played on TV. In addition to Maltese Falcon (1941), here are a couple of his more memorable roles, at least for me:
Harry's War (1981) .... Sgt. Billy
1941 (1979) (as Elisha Cook) .... The Patron (Dexter)
House on Haunted Hill (1959) (as Elisha Cook) .... Watson Pritchard
Shane (1953) .... Frank 'Stonewall' Torrey
Maltese Falcon, The (1941) .... Wilmer Cook
That's citing only 5 movies from a body of work that spans 57 years and includes over 200 character roles. When Jack Palance killed his character in Shane you really wanted to see Shane blow him away. It was the perfect set up for a movie showdown.
Harry's War is unknown to most people. It's an anti IRS comedy from 1981 that has been ruthlessly suppressed (according to the stories I've heard) by the IRS. As I hear it virtually everyone involved has had their taxes audited every year since the film came out. Not a chance anyone will put it on TV or on DVD!
You're the only other peron I've met who's seen that movie.
BTW, you may be enough of a lover of strange movies to get this one:
"Don't get officious. You're not yourself when you're officious. That is the curse of a government job."