At one point all women were required to cover their head for church. In general unmarried women were not obligated to cover their hair if they were not in church. Married women during the medieval era often cover their hair until the the custom completely died off sometime during the Renaissances depending on the country. A common headcovering for married women was the wimple, this is where nun's headdress derived from. King James version of Isaiah 3:22 even stated The changeable suits of apparel, and the mantles, and the wimples, and the crisping pins. Around the mid-1400's transparent wimple appeared to show hairstyle underneath. After the 1500's there were many more varieties of head covering available. For example the book Inside the Wardrobe of Anne Boleyn described unmarried Anne Boleyn loved to wear her long hair loose, but after she married Henry VIII she was obliged to cover her hair. She often wore French hood that didn't cover all the hair completely. After Elizabethan era there seem to be no major difference between headwear of married and unmarried women in Western Europe except for public worship in some denominations.

In Eastern Christian denominations women were often required to cover their head in church (even today).

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Catholic women generally wore chapel veils, or mantillas, until around the time of the Second Vatican Council.

I'm not sure if it was big in the United States because the American Catholic Church was dominated by the Irish and there were things they didn't do because outward displays of their faith was basically illegal in Ireland until the mid-19th century.

In some places it is still customary for women to wear chapel veils. I believe it's common in South Korea, for example.