Women voters had been skewing politics toward more government largesse long before the "Great Society," in fact even before the Nineteenth Amendment was ratified c. 1920.
I saw a study that evaluated state spending data from the period just before the Nineteenth Amendment. States that allowed women's suffrage had significantly greater increases in state spending in the years immediately following the start of women voting, when compared to states that did not allow women's suffrage in a comparable time frame. Conclusion: Women in general supported more government spending than men and put more pressure on government in that direction.
That pattern continued at the federal level, but with a lag period. Harding and Coolidge, the first two presidents elected with a significant number of female voters, were fiscal conservatives. But then came Hoover, who introduced some new government programs, followed by Franklin D. Roosevelt, who could be viewed as "Hoover on steroids" in terms of government expansion.
The question was when women voters began to outnumber men voters.