Headrests really didn't start becoming commonplace until the late 1960s / early 1970s. They were made mandatory in the US only on January 1, 1969.

http://www.webcitation.org/62vMMUPPX

This post-dates the hatless trend by about a decade, though it certainly wouldn't have helped the haberdasher's trade.

Good points. Caused me to do some additional research (beyond my personal experience with hats).

Patents for headrests were granted in 1921, 1930 and 1950. [1] According to some sources, they began to appear in cars in the 50s and 60s [2], which coincides with (and slightly pre-dates) the JFK theory.

1: http://en.wikipedia.org/wiki/Head_restraint 2: http://www.ehow.com/facts_7597635_use-headrest.html