Mail art by George Henry Edwards, 1900
Whenever I read that mail art began “in the 1960s,” I roll my eyes. Mail art has been around as long as creative people have been mailing, and wonderful examples from the 19th and 20th century abound. 
Mail art, 1873
Mail art by Hugh Rose, 1910
Mail art by Frederick Charles Tolhurst, 1918
Mail art, 1926
Granted, in the 1960s artists in New York City, rebelling against the “elite” art establishment and its rules, began sharing art through the mail with “no rules.”  But to say they were the first mail artists and that only their work can be called “mail art” is an over-reach. Like trying to lay claim to “tea cup.” Perhaps they did think they were the first mail artists; after all, there was no Internet then where they could search for earlier examples.
Mark Bloch writes, “Ed Plunkett, the man credited with coining the phrase New York Correspondence School, once said that postal art probably got its start the day Cleopatra had herself wrapped in a blanket and delivered to Caesar. As long as there has been mail, there have been people embellishing it in various ways, both physically and, more importantly, perhaps, theoretically. Anyway, as long as there have been artists and there has been post, there has been postal art.” 
So let us call the mid-century New York version “Correspondence Art.” Or spell it as “Correspondance Art,” in which Ray Johnson alluded to the dance of corresponding and exchanging art. In Fluke: The Mail Art Issue, John Held notes, “I consider Mail Art a movement and I capitalize it. Like Dada Art or Fluxus Art… A lot of people don’t see it that way; they see it uncapitalized like ‘painting’ or ‘collage.'”
Given that distinction, I can see “Mail Art” as a movement that originated among artists in the 1950s, but “mail art” is much older and belongs to the world: art created by an individual to be mailed and experienced by everyone who sees it on its way to the addressee.
Maybe Mail Art
Artist’s postcard, Henri Cassiers, 1902
Artist’s postcard, Ichijo Narumi, 1906
Artists’ postcards, i.e., works of art meant to be posted, but printed in multiple quantities, fall into a gray area. The sender is not usually the artist, and although these postcards are certainly art, and are mailed, their status as mail art is open to debate. (For $5, I can take either side.)
Very notable in the genre was the Collection des Cent, created by Émile Gréningaire, a Parisian watercolor painter and illustrator. In 1900, he issued an open invitation to artists to create artworks especially for postcards. His goal was to issue a series of packets holding 10 postcards each, publishing a packet every 15 days. The first cover was marketed on November 1, 1901, and the ninth and last in May of 1903. 
The postal art of Donald Evans – his tiny stamps from imaginary countries – was amazing in every way, but never meant to be actually mailed; certainly it is postal art, but perhaps not mail art. 
The postage stamp paintings of Molly Rausch, so fabulous. But, not posted, not quite mail art.
Letter from Edward Ardizzone
Illustrated letters, sent inside the envelope, also lurk in the gray area. 
Not Mail Art
Pictorial envelopes, printed for business use and advertising, can be artistic, but are not mail art.
Emily Dickinson’s envelope poems, written on scraps of envelopes from letters she received, are certainly art, but not mail art. 
The art of Amy Rice on vintage envelopes, lovely art, but not mail art.
The Mona Lisa, if wrapped up in brown paper and mailed to Rome, would be art in the mail, but not mail art.
Stamps created by artists, both on stamp sheets and individually posted, certainly live under the umbrella of mail art, but are a subject for another day.
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 British Pictorial Envelopes of the 19th Century (1984) by Ritchie Bodily, Christ Jarvis & Charles Hahn
World War II Envelope Art of Cecile Cowdery (1992) by Robin Berg
Hand Illustrated Postal Envelopes (1996) by David Swales
Les Plus Belles Enveloppes Illustrees de 1750 a nos jours (2000) by Pierre-Stephane Proust
The Louis Grunin Collection: A Philatelic Art Gallery (2010), auction catalog from the Robert A. Siegel Auction Galleries, Inc.
On the Web:
Search “hand illustrated covers” on Google.
 No rules, except their own. They decried the rules of gallery exhibitions but listed “considerations.” They mocked bureaucracy but were dependent upon the post office. They protested the closed nature of the art world, yet insisted that mail art was mail art only if it was exchanged within the International Mail Art Network (IMAN). See Anna Banana interview by Zora von Burden in Women of the Underground: Art (2012). See also Small Scale Subversion: Mail Art & Artistamps (2015) by John Held Jr., the Boswell of this aspect of the Mail Art world.
 Album de Cartes Postales (1960), afterword by C. Lauterbach & A. Jacovsky
Art Nouveau Postcards: The Posterists’ Postcards (1977) by Alain Weill
Art of the Japanese Postcard (2004) Museum of Fine Arts, Boston
 The World of Donald Evans (1980) with text by Willy Eisenhart
 Les Plus Belles Lettres Illustrees (1998) by Roselyne de Ayala & Jean-Pierre Gueno
More Than Words: Illustrated Letters from the Smithsonian’s Archives of American Art (2005) by Liza Kirwin
 Emily Dickinson: Envelope Poems (2016) edited by Christine Burgin