JFK wanted to appear forward-looking, the new frontier and so on.
Fedoras, nearly required in the 1950s, suddenly dropped out of fashion in the 1960s.
To wear a fedora in 1963 would have signaled fuddi-duddiness.
JFK declined, even for a moment, to wear a Stetson, while in Dallas.
Having lived in Texas for a couple of years, I think a photo of JFK in a Stetson would have been a smart PR move.
I don't know why JFK did not doff the Stetson, at least for a photo shoot, in deference to Texas customs. JFK usually knew instinctively what was good PR. But perhaps Texas was terra incognita to JFK.