depends on what type of work and where it is.
I have worked with and around engineers at Charleston Naval Shipyard(best job I ever had), designing bottling, packaging, and canning machinery, and Cummins Marine.
the majority of engineers at CNSY wore hats when in the office due to hardhat hair, even if they didn't have any hair.
in the bottling facility almost all wore hats on the floor and more than a few in their offices unless they had an important meeting.
at Cummins when you were out on a boat or in the testing or manufacturing area it was common to see a lot of hats. some of the senior engineers were known for their hats.
if you're always inside or don't deal with the public you'll probably only see people wear one to and from work. if your the type of engineer that gets to go outside, into manufacturing areas, and such hats will probably be fine.