"Sidewise in Time" was among the first science fiction stories about parallel universes. In 1903 H. G. Wells wrote "A Modern Utopia" in which people from our timeline were shown traveling to another, but Wells used this mainly as a literary device to present his speculations of a perfect society. Leinster's story, conversely, introduced the concept to the pulp science fiction readership, bringing about the creation of one of the field's subgenres. L. Sprague de Camp's 1940 story "The Wheels of If" followed a single man as he was involuntarily transported through a series of alternate timelines. H. Beam Piper's paratime series (1948–1965) postulated the existence of a civilization that could travel at will across the timelines, a theme echoed in Larry Niven's "All the Myriad Ways" (1968), Frederik Pohl's The Coming of the Quantum Cats (1986), and Harry Turtledove's Crosstime Traffic series (2003–2008). Other stories dealing with travel to parallel timelines include Isaac Asimov's "Living Space" (1956), Keith Laumer's Imperium series (1962–1990), Jack Vance's "Rumfuddle" (1973), and Jack L. Chalker's G.O.D. Inc trilogy (1987–1989). Lawrence Watt-Evans' story "Storm Trooper" (1992) is set in world whose inhabitants, like those of "Sidewise in Time", must cope with the sudden appearance of sections of other timelines. Gordon R. Dickson's Time Storm (1977) depicts an Earth ravaged by a cosmic storm that randomly changes the historical periods of local regions, much like "Sidewise in Time". The anime series Orguss took "Sidewise in Time" as one of its inspirations, and showed the world caught in a trap of constantly changing territories of alternate-Earths.
The setting of Fred Hoyle's October the First Is Too Late (1966) is similar to that of Leinster's story, except that the segments of Earth which are brought together and interact with each other are from different historical periods, rather than from different parallel histories.
In his comments on the story in Before the Golden Age, Isaac Asimov writes that "Sidewise in Time" had a long-term effect on his thinking. "It always made me conscious of the 'ifs' in history, and this showed up not only in my science fiction, as in 'The Red Queen's Race', but in my serious books on history as well. I also used the alternate-history theme, in enormous complexity, in my novel The End of Eternity."
Themes touched upon by Leinster would be taken up at greater length by others: Confederate victory in the American Civil War, a Roman Empire which never fell, enduring Viking colonization of America, Russia keeping its 19th-century colonies in Pacific America, Chinese colonists finding their way to America.
The idea of a scholar using a cataclysmic event to make himself the ruler of primitive people was taken up by S. M. Stirling in the Nantucket and Emberverse series, where the main villains do this repeatedly.