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During the 1940s and into the 1950s or 60s, all men, when out in the street wore a dark beige or grey hat that was typical of the time. It was called a 'trilby'.
So common was this hat-wearing that it was always expected that men would have somewhere to place their hats when they came indoors. In homes, this was the hall stand, and in offices and restaurants it was a hat stand of the sort shown in the photo, (In wealthy houses in previous years, the butler was always expected to take visitors' hats.
It was common practice at the time for men to touch these hats in a type of salute whenever they passed a woman acquaintance. This must have been a leftover from previous generations when men doffed their hats to ladies, sweeping into a low bow while holding the hat in one hand.
By the mid 1900s, though, doffing had degenerated into a mere token. Nevertheless it was a normality as all the men seemed to do it.
Incidentally my father would never have referred to any female he knew as a 'woman'. She was always a 'lady'.
The practice of men touching hats to women, probably went out of fashion at the same time as trilby hats went out of fashion.