Kleenex was apparently not originally intended for nose-blowing. From “It’s Spreading,” Jill Lepore’s excellent 2009 New Yorker article about the media feeding frenzy that created the Parrot Flu scare of the 1929-30:

“By the twenties, Americans, and especially housewives, lived in fear of germs. Not only did newspapers and magazines run almost daily stories about newly discovered germs like undulant fever but their pages were filled with advertisements for hygiene products, like Listerine (first sold over the counter in 1914 and, in many ways, the granddaddy of Purell), Lysol (marketed, in 1918, as an anti-flu measure), Kotex (‘feminine hygiene,’ the first menstrual pad, introduced in 1920, a postwar conversion of a surgical dressing developed by Kimberly-Clark), Cellophane (1923), and Kleenex (1924; another Kimberly-Clark product, sold as a towel for removing makeup until a consumer survey revealed that people were using it to blow their noses).” (Thanks Longform.)

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Baby ogre sells Kleenex in Japan, 1986: