Sounds more like a Top Hat that has been folded down.

First known use of the word was in 1940.  However, 

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Legend has it that the term flat-hat originated with an incident back in the days of barnstormer in which a pedestrian's hat was crushed by a low-flying airplane. According to one version of the tale, the reckless pilot was subsequently required to purchase a new hat for the hapless pedestrian. (Merriam-Webster)

Barnstormers not the only pilots that flat-hat.  In the 1920's recycled WWI planes were used to do crop dusting for insects.  Fun Fact:  One of the largest crop dusting companies in the US, eventually became Delta Airlines.