The trouble is that we don't really know - I haven't seen or heard of any primary sources that discuss how to hide your shortened hair. It's worth mentioning that very few women would go for a drastic haircut! During this period, it was really only done for a serious illness; earlier in your period, because it was believed to actually assist in her recuperation, and by the twentieth century, more out of habit.

However, I can tell you what one woman from the end of the period you've asked about did! Irene Castle was a huge celebrity of the 1910s, the ragtime-and-tango-era. Mr. and Mrs. Vernon Castle made pots of money dancing together on stages across the world, as well as their tie-in merchandising deals and their dance halls and schools. In May 1914, once the couple got back from their "Whirlwind Tour" across America (culminating in a show at Madison Square Garden), Irene was rushed to the hospital for an appendectomy; while she recovered enough by July to appear in public again, she would be back in the hospital to deal with complications resulting from the surgery in January of 1915.

Before her first hospital stay, Irene cut her hair short, probably due as much to custom as to her stated reasoning:

I never liked having anyone comb my hair, so, to assure as little combing as possible, I cut it all off. I say all; it never fell much below my shoulders.

After she left, she hid her hair as much as possible from the few people that she saw in her public role by wearing a turban and pushing her hair up into it so that she appeared to have a hidden updo. But, according to her account, her friends who saw her in private without the headgear found her hairstyle adorable. They persuaded her to go out and about with her hair unhidden, and ...

so one night when we were going to town to dinner I wore it down, and in order to keep it in place wrapped a flat seed-pearl necklace around my forehead—which was, I think, the beginning of what they afterwards called the “Castle Band.”

The "Castle bob" became a sensation and a much-copied trend, whether through an actual cut or a rolled-up imitation, as attested in newspaper references in late 1914 and through 1915 and 1916 - the 1915 Illustrated Milliner called it "extremely youthful" and noted that it was "rapidly spreading through the country". (Although we have to take that with a grain of salt, since the bob would continue to be controversial into the 1920s.) And no wonder! Cecil Beaton would later say that she was "one of the most remarkable fashion figures the world has known."

What's really interesting to me is that Irene Castle was well-known as the originator of the bob haircut for decades. The Fred Astaire/Ginger Rogers film The Story of Vernon and Irene Castle (1939) devotes a few moments to the popularization of the bob in such a way that it's clearly aimed as a nostalgia trip at adult audience members of the time. It's not until sometime after Coco Chanel returned to prominence following her post-WWII preventative exile that she was able to retell the story with herself as the bob's originator.

The NYPL Digital Collections has a good number of publicity shots of Irene Castle with her signature hairstyle.