In 1930, Odell Shepard, English professor at Trinity College (Hartford, Connecticut), published a book called The Lore of the Unicorn, an attempt at an exhaustive history of the unicorn in literature and historical writings from the fourth century BCE to the nineteenth century CE. This book, which has been reissued in the 1970s, 1980s, and 1990s, contains the word alicorn/alicorns over 150 times. On page 101, Shepard explains the origin of the word:
In order to avoid repeated cacophany [sic] I shall use the word "alicorn" to mean "unicorn's horn" wherever it seems convenient to do so in the following chapters. This is not quite a neologism; it is an adoption of the Italian word alicorno.
It's possible that Shepard's work may have (re-)kindled interest in unicorns and popularized the word "alicorn." Three years after Shepard's book came out, Alexander Laing published a novel called The Sea Witch about the experiences of the captain of a clipper ship in the mid-nineteenth century. This novel uses the term alicorn/alicorns about 25 times. It has a chapter called "The Narwhal's Tusk" in which Laing compares the narwhal's single horn to "alicorns — horns of the four-footed monoceros."