Possibly, but 60's cars weren't much more advance than the 50's, except for new optional items. As the parents deferred to their Boomer kids antics, the hippies and others would mostly pool their resources for an used car. I did it with friends, although was never a hippie.
The exception would be the rich who bought new cars for their spoiled over-indulged kids, mostly on the way to a college where they took their "anything goes" beliefs with them that started the dumbing down of academia. Those who couldn't get a car, hitch-hiked. That was big in California. Are you talking about other means like buses or rail or what?
The main drivers of increased mobility were the building of the interstate highway system, initiated under Eisenhower—they built it, and millions came—and the postwar affluence that enabled families to own more than one car, including cars for teenagers.