The question is in the title, but to expand on that, it seems to me like opening such a vast, and previously unaccounted for, demographic after all the wheels of an election had started turning, could really throw a wrench in how a presidential candidate might operate for the rest of the campaign, how did the candidates at the time handle that?
It's important to clarify that the expansion of women's votes nationally was significant, but not as significant as one might assume, given that a number of states had already legalized women's voting earlier, most notably the nation's largest state, New York, in 1917. By 1920, most of the western US already had women's suffrage, as well as some states outside the West, like Minnesota, Michigan, and Rhode Island.
States that had suffrage already prior to the 19th Amendment were: Wyoming, Utah, Idaho, Washington, California, Arizona, Kansas, Nevada, Montana, Rhode Island, Oklahoma, New York, South Dakota, and Michigan, along with the Alaska Territory. (In addition, Illinois had Presidential and city-level representation for women, but not for statewide races)
One of the main ways women's suffrage passed to begin with was the pressure women voters in states that had already granted women's suffrage put on their politicians to expand it elsewhere. The path to 2/3rds majorities necessary to pass an Amendment became far easier when the 1/10th of the House of Representatives from New York State suddenly had half their electorate be women. The process of adjusting to large-scale women's voting had already begun well before the Amendment had passed.
One region where the effect was more dramatic was the South. Not a single state that had women's suffrage before the 19th was in the South, meaning that the Amendment had a far more dramatic effect on that section of the country in terms of enfranchisement.
The effect on the raw vote total was dramatic. According to Dave Leip's US Election Atlas, in the 1916 Presidential election, 18,541,312 votes were cast overall. In 1920, that number soared to 26,788,225 votes cast, an increase of nearly half again what it had been before. In 1916, the winning candidate, Woodrow Wilson, got 9,130,861 votes, 49% of the total. In 1920, the losing candidate, James Cox, got a nearly-identical 9,140,256 votes, but that was a mere 34% of the total vote, setting Cox up on the list of one of the all-time lowest vote percentages of a major party candidate.