Women were granted the right to vote in Wyoming Territory in 1869 (it's one of the main reasons that one of Wyoming's state slogans is "the equality state"). But, clearly what Hillary was referencing was the national right to vote, which came later in 1920. Congress passed the amendment in June of 1919 but it was not ratified into law until August 26, 1920 (of course, famously Tennessee was the state that tipped the balance towards successful ratification). So, from the date of the first women's suffrage convention in Seneca Falls, NY in 1848 until 1920... it was a 72-year long fight to gain women the right to vote in this country. This of course, does not count the struggles towards this idea that happened prior to 1848.
You're right, in the early years of the movement it was a state-by-state struggle and strategy for the suffrage movement. The strategy shifted around 1913 to be a huge, full-fledged national push... leading to the 19th amendment. The young suffragist Alice Paul, along with a more established suffragist named Carrie Chapman Catt were the two who really saw the movement through to the end in 1920. Susan B. Anthony, sadly never saw the national goal achieved.
When Susan B. Anthony died in 1906 there were 4 states where women could vote... all in the west: Colorado, Wyoming, Utah and Idaho. She was buried with a flag with only four stars on it. And, famously, Anthony was arrested in 1872 for the act of voting and she wrote the language that is now the text for the 19th amendment way back in the 1870s.
I co-curated an exhibition on women's suffrage a while back when I was still in graduate school... so this is a particular interest of mine.
Here's a fairly good online timeline of the history of the women's suffrage movement in the U.S. It's truly an incredibly history. People nowadays forget what a huge movement and struggle it was.
http://dpsinfo.com/women/history/timeline.html