When men raised their hats to women
Men's hats in the mid-1900s
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, particularly in previous years, the butler was always expected to take visitors' hats.
Raising hats to women
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 swept into a low bow while holding the hat in one hand. This was known as doffing the hat.
By the mid 1900s, though, doffing had degenerated into the mere token of simply touching their hats when they saw a woman they knew. Nevertheless it was still 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 hats for men went out of fashion.