The one way it would work is to have a pop top or elastic roof. W.C. Fields had a car like that in the 1933 movie International House (hilarious movie and pre-code, too).
The roof was rubber and his head would push it up a few inches. He'd put his top hat on top of the bulge in the roof. By they way, that's Burns and Allen on the roof and hood and, yes, that is Bella Lugosi pushing from the back.
The site I link to above that talks about the movie does get at least one thing wrong. They say the film was set in Wu Wu, China. It's really set in Wu Hu, China. That's important to the first Fields gag in the film. He's an adverturer / aviator flying from Kansas City, Kansas to Kansas City, Missouri in his autogyro and he's lost. He lands his aircraft on the roof garden of the International House Hotel in Wu Hu China (he's really lost and the movie is that silly). He leans out and asks this very attractive (by the standards of the time) and scantily clad woman "Where am I?" She replies "Wu Hu." He leers at her and says "I'll talk to you later" then turns to the supercilious manager (the guy in the black suit sitting on the front of the hood above) and repeats the question. When the manager also answers "Wu Hu" Fields pulls the flower out of his lapel, throws it away and mutters "Don't let the posie fool ya." As I said, pre-code.