Drew, my favorite movie of your dad's was The Asphalt Jungle (1950). His character was named Dix. Jean Hagen played his wife. Marilyn Monroe also had a bit part. It's very suspenseful. I'm sure there are lots of phones in it too. I posted a neat picture from that movie below with all the major stars looking down through a glass table. Your dad is bottom right. Jean Hagen is next to him on the left and Louis Calhern is next to her. Monroe is upside down on top. She played Calhern's girlfriend. The movie is going to be on Turner Classic Movies on Friday March 26th at 2:30pm Eastern, so I may record it and capture some of the phones.
I haven't seen Dr. Strangelove all the way through, but I agree about the phone being pivotal in the movie. I found a bunch of stills from it all over the net and they're posted below. The first one is your dad with a long cigar, in case anyone doesn't know who we're referring to here. I left the title as is. "Essence" fits it well, don't you think?
Next is Peter Sellers on the phone. It looks like a British phone to me. Next is Sellers as a different character, on a G3 handset. Next is an odd one, from Getty Images, where they are filming Strangelove and the girl in the bathing suit is on the phone. Next is another shot of the girl and you can see the phone in the background.
I have a couple more shots if anyone is interested. Another interesting bit of trivia... Peter Sellers and my dad were born on the same day in the same year, September 8, 1925. They were nothing alike.