Touch tone phones (DTMF) were first deployed at scale in the US in 1963, but they would have been in development at the Bell System for quite a while before that. Possibly also some limited deployments. This ad from 1956 teases a future without rotary dials.
DTMF also led to the rise of Blue Boxes and Phreaking.
By the time the real DTMF phones appeared, they did get the the keypad into the base, so the handset part was still slim. Not sure if that was all of them, or just some models. But it looked like so:
DTMF - dial tone, motherfucker?
DTMF - dual-tone modulated fone
Dual tone multiple frequency
That’s the version Sam Jackson gets.
Chunky and satisfying, like banana bread
Those things made an impressive noise when Russell Crowe used them