I’ll nominate one of my favorites in this genre: Harry Nilsson’s “Good Old Desk.”
My old desk
Does an arabesque
In the morning when I first arrive . . .
I guess a lot of Nilsson stuff would qualify. Shame he’s gone. 
Not to mention her “The Sensual World” (and it IS!)- a very sexy song inspired by James Joyce’s ULYSSES!
Hawkwind has a song called “Orgone Accumulator”. You can probably guess what it’s about. (And I suspect the majority of Hawkwind’s catalog would be appropriate for this thread.)
Pressurehed put out an album called Explaining the Unexplained–each of the songs is about some Fortean phenomenon like Bigfoot, UFOs, Mokele-Mbembe, mammoths preserved in ice, and so forth.
Sinead O’Connor has a song called I think “Famine”, a kinda rap/polemic about the Irish Famine that uses lines from Eleanor Rigby. It’s fairly non-traditional subject matter.
Michael Flanders and Donald Swann covered various kinds of animals (including cockerels in Greek and camels in Russian), trams, buses, closed down railway stations, nuclear war, the ascent of Mount Everest and the Second Law of Thermodynamics in their songs. Among other topics.
And there’s always Tom Lehrer …
I refuse to concede that a person is well-read just because they’ve read “Wuthering Heights.”
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True, but how about, as FriarTed points out, Ulysses?
Kate has many songs based on or inspired by books and movies. Off the top of my head, there’s “The Infant Kiss” based on the film The Innocents, which in turn is based on The Turn of the Screw by Henry James; “Cloudbusting” which is based on the Peter Reich book A Book of Dreams, about his father Wilhelm Reich; “Get Out of My House” which is inspired by the book (not the movie) The Shining, by Stephen King; the song I mentioned, “Pull Out The Pin,” is inspired by a BBC documentary; “Delius,” inspired by a BBC movie about the composer; “The Red Shoes” inspired by the film; more I’m sure but we’re off to see some movies.
Speaking of Wilhelm Reich…there’s, of course, Kate Bush’s Cloudbusting, as EQuipoise noted (But I caught the connection, so I’m re-mentioning it).
Some 10000 Maniacs songs are like this …
I’m Not The Man, about wrongful capital punishment.
Gold Rush Brides, about pioneer women
*Hey, Jack Kerouac *, about…well, Jack Kerouac.
Eat For Two, about pregnancy (see This Woman’s Work, K. Bush)
And Pixies are good for some obscure topics - salaryman suicides, mars, aliens, U.Mass., Lost Continents, Biblical incest, Dali movies…
Primus also has a song about Jack Kerouac, simply called “Jack Kerouac.”
One of my favorite bands is Splashdown, who have quite unfortunately disbanded. (All their music is available for download on the internet, though, with full approval of the band members.) A lot of their music is on non-traditional subjects, from “A Charming Spell”, which is about paganism/Wicca, to “Elvis Sunday” which is about saying goodbye to childhood (I think), to “Karma Slave” which is about karma. Ooh, and “Mayan Pilot” is inspired by those images on Mayan buildings that look like astronauts.
This is why I listen to a lot of folk music (real folk music with “trad.” as the author)–because I get tired of songs about love affairs and partying, and want to hear about a battle or a sea voyage or a hanging, or meeting the Devil on the road, or Napoleon’s exile, or Scottish politics. Bit of irony there, that traditional music so often deals with what the OP calls “non-traditional” topics.
The Pogues did plenty of nontraditional stuff in their time, some “trad.” (“Jesse James”, “Greenland Whale Fisheries”), some of it their own (“Smell of Petroleum”, “Haunting”, “Bottle of Smoke”, “Thousands are Sailing”) and some by other writers (Dick Gaughan’s “Navigator”, Eric Bogle’s “And the Band Played Waltzing Matilda”).
The Decemberists write about some pretty non-traditional subjects. A few examples:
“The Mariner’s Revenge” - the tale of a sailor who takes revenge on the cad who disgraced his mother (but is not his dad), after both men have been swallowed by a whale
“Eli the Barrow Boy” - a poor boy dreams of a better life for himself and his beloved, but she dies and he commits suicide soon after
“The Sporting Life” - a high school football player commits an error during a game and realizes that the people in his life love him for his athletic skills, not for himself
It should be noted that dance diva Amber also did a song consisting partially of Molly’s soliloquy from Ulysses.
The song Too High for the Supermarket is about grocery shopping while stoned.
Wilco’s Spiders (Kidsmoke) is about spiders. and tax returns. I think. I’ve never been able to tease out the meaning, but I’m pretty sure whatever it is, it’s not traditional.
How about Jim Stafford’s Swamp Witch? Or Spiders and Snakes? Or My Girl Bill?
I’m the first person to mention Kinky Friedman? He wrote a delightful song about racism (They Ain’t Making Jews Like Jesus Anymore), about discrimination (We Reserve the Right to Refuse Service to You) and the best damn song about the Holocaust ever written (Ride Em Jewboy).
And musical theatre has a few, including (to quote Mr Burns) “a musical about the common pussycat or the king of Siam.” But nothing tops originality for writing an entire musical about the wife of an Argentinian President.
Off the top of my head I can only think of a couple songs by them about romantic love or breaking up (Jezebel comes to mind.) Then again, on The Wishing Chair they had some traditional or traditional-sounding tunes, in a folk and/or celtic sense rather than trad blues rock sense.
On both albums of Hope Chest, the topics are nearly all way out there (children’s programs, science, artists, multiple personalities: ) the ersatz Rasta songs and happily-sung anti-war ditties are the most familiar topics there.
And nearly all of these would make great band names 
The Sniper by Harry Chapin, a song about Texas Tower sniper Charles Whitman.
Dead Skunk (In The Middle Of The Road) by Loudon Wainwright III, self explanatory.
Alice’s Restaurant by Arlo Guthrie, a song a littering ticket (with 27 8x10 color glossy photographs, with circles and arrows and a paragraph on the back of each one.
Charlie’s classmate Kinky wrote one too: The Ballad of Charles “Texas Tower” Whitman