Yeah, that's entirely possible. I have no idea how those car radios functioned in the 1930s, but the pictures I've found online of vintage car radios at least indicate that they had a volume knob, a tuning knob, and some means of turning the thing on and off. In some pics I've seen, it even appears the radio had a little key that fit into it… I'm assuming that was a key for turning the radio on-off (looked too small to be the car's ignition key). As a matter of fact, you can even see said key inserted in the radio of the 1936 Packard, down below the volume & tuner knobs. I magnified the radio a bit and took a screen shot where you can see it better:

Often in old cars, the on-off and volume were on the same knob.
From the off position, one would start turning the knob and feel a bit of a click, turning the radio on, and then as one continued to turn the knob the volume would increase.
Changing the tuning probably would have worked. Although it was an AM radio.

Guess the plot required it.
My password is password.