Looking at driver's licenses held by women and considering that manufacturers were aiming certain car models specifically at women, female drivers were not uncommon or unusual in the US in the 1950s and 1960s, or even before that. As you allude to in your question, this is in marked contrast to the UK.
In the 1950s in the US, about half of adult women had a driver's license. In 1960, with 39% (or 34 million) of registered drivers being women (compared to 53.2 million men), around 55% of women had a license. By 1965, 40.8% (or 40.2 million) of registered drivers were women (compared to 58.3 million men). In the 1960s in the UK, on the other hand, men were more than twice as likely as women to have a driving licence.
There was certainly nothing unusual about women being behind the car wheel. Even in the early days of the automobile, women were making their mark as drivers - see Alice Ramsey's 1909 trip across America, for example.
As early as 1916,
The Girl Scouts initiated a “Automobling Badge” for which girls had to
demonstrate driving skill, auto mechanics, and first aid skills.
The US car industry recognized the importance of women drivers from at least the 1920s:
Beginning in the 1920’s and 1930’s many major automobile manufacturers
recognized the growing trend of women driving for fun and necessity.
They began to gear their print ad campaigns to women...
Source: Antique Automobile Club of America Museum, 'Women's Automotive History Highlights'
Even in 1926, Ford was advertising its Model T Tudor Sedan for women, claiming that
Inquiries reveal why women are so highly enthusiastic about the
present Ford car
In the same year, another advertisement says the Model T Coupe
is an ideal car for women's personal use
In 1954, Nash motors specifically targeted women with their Nash Metropolitan, described as a “commuter/shopping car”. This came a little before the Dodge La Femme, shown below at 1956 Chicago Auto Show.
"In this close-up view of a Custom Royal La
Femme 4-door hardtop, at the Dodge exhibit
space, the car is on a raised platform with a
rotating floor. Two female models are next to
the vehicle, while a spokesman at left points to
the car. Note the open umbrella, which was part
of the accessory package that came with the La
Femme model--unabashedly aimed at women." Text and Image source.
As today, the level of car ownership and usage was not evenly distributed among social groups, and it also varied among urban, suburban and rural households. Already in the late 1940s,
Driving was becoming a necessity for most suburban housewives ...
because of the location of postwar housing and of retail and
industrial developments.... traffic engineering encouraged the growth
of suburban communities that favored safety and privacy but were
unfriendly to walking and to transit systems (Jones 2008, 120-2).
Already in the 1950s, strip developments on roads and then shopping
malls were constructed to centralize retail outlets. Supermarkets
expanded to embrace a full range of food and then to add other
household commodities such as stationery, toiletries, and white goods.
Even if these centers were accessible by bus, women could not carry
large loads, especially if accompanied by young children. They could
not function effectively as household managers if they were stranded
in spread-out suburbia.
Source: Margaret Walsh, 'Gender and Automobility: Selling Cars to American Women after the Second World War'. In Journal of Macromarketing (August, 2010)
Comparing the US and the UK in the 1960s
In the US in 1960, approximately 55% of adult women had a driver's license; his percentage had increased slightly by 1965 (calculated from data here and here). In contrast, in the UK in the 1960s, the figure for women was less than 20% of adult women. Even in 1975, women held less than 30% of driving licences (i.e. 70% for men) in the UK.