This went out in my lifetime. As a small child, it was very 'statementy' for a bloke to go out without a hat. Downtown Salford where I lived was not a place to be different unless you were weell hard, so most blokes wore hats, usually flat caps, again so as not to stand out.
Workmen wore hats that were usually linked to their trade - just think of Norman Wisdom in his numerous roles as this illustrates the point well.
In my memory - not a proven established fact - men's hats seemed to go alongside changes in hairstyles, from the brylcreemed side-parted short back n sides, to gradual creeping in of longer lengths and the abandoning of the grease. I would put a date of about 1962-3, guessing, of the tipping point going from hats/ brylcreem to bare heads. Older men held on to the brylcreem longer, but the floppy-haired look for older teens and young men seems to have passed from Aldermaston Lefties to mainstream by ths time.
It must be that this ties in with the explosion of youth culture and spending power. As with all things, you can expect this to be a trend that starts in the affluent south and more slowly spreads 'oop north'. The big exception, and again an undoubted tipping point, was the merseybeat era and especially the arrival of the Beatles.
This is also about the time that expectations about how you 'should' behave as an adult shifted and relaxed - looking back at the fifties and early 60's, people didn't half look old at an early age. Think of the contrast between Ena Sharples and whatever character Babs Windsor plays in 'stenders and you'll see what I mean.