When you got to a new town, I seem to recall pulling a booklet out of the glove compartment, finding the right page, finding the right frequency for the station you're after, and tuning manually to that frequency.
"Punch a button" was only an option once you'd already saved that preset - no difference to the later digital version.
Even some 70's era stereos had that capability -- my grandparents had a mid 1970's Cadillac with an analog tuner, but it also had a button you could press and you could watch the tuning needle scan until it found a a station.
I was looking for a link to one, and it turns out that even some 1950's radios had a "wonderbar" signal seeker:
"Punch a button" was only an option once you'd already saved that preset - no difference to the later digital version.