ומדר"נ בר יצחק נמי אין מזל לישראל דאימיה דר"נ בר יצחק אמרי לה כלדאי בריך גנבא הוה לא שבקתיה גלויי רישיה אמרה ליה כסי רישיך כי היכי דתיהוו עלך אימתא דשמיא ובעי רחמי לא הוה ידע אמאי קאמרה ליה יומא חד יתיב קא גריס תותי דיקלא נפל גלימא מעילויה רישיה דלי עיניה חזא לדיקלא אלמיה יצריה סליק פסקיה לקיבורא בשיניהShabbat 156b
Translation: And from Rabbi Natan bar Yitzhak,we can also learn that there is no fortune for Israel: A Chalden said to the mother of Rabbi Natan bar Yitzhak "Your son will be a thief". She did not permit him to reveal his head. She said to him- "Cover your head, so that the fear of heaven will be upon you, and pray for mercy". He did not know why she said this to him. One day, he sat and was learning under a palm tree. His cloak fell off his head. He lifted his eyes and saw the palm tree. His [evil] inclination became strong, and he cut off a bunch of dates with his teeth.
This Talmudic tale tells us that wearing a covering of some sort gives one the fear of heaven, and even suggests that it has the power to prevent us from sinning, even or perhaps particularly in ways that are natural for us, individually. In this manner it is a preventative measure, and a step toward being our best selves, rather than our worst ones.
It also does not, at this point, presume that this is something that most people do, nor that there is any general requirement to do so. On the other hand, it makes the practice look very attractive- a benefit without any detriment. It's a way of helping ourselves to be good.
It also does not say anything about what sort of head-covering is involved- Rabbi Natan bar Yitzhak used his cloak/outerwear as his head-covering. So any hat, scarf, turban, etc ought to work. And as it is a story and not law, there are no suggestions of any sort about how much of the head needs to be covered for this preventative to work. It doesn't even seem to have anything to do with personal awareness- our protagonist has no idea why he covers his head normally, or that it has fallen off in this instance.
Therefore, it would seem that if the community at large were to adopt this practice (as of course, has happened, to a great extent), gender might not play a significant role. There's nothing strongly gendered about this story, as far as I can tell...