Came in to post this (I thought that song was pretty obscure!)
Instead I will add March 1, from the less-obscure “Casimir Pulaski Day” by Sufjan Stevens:
“In the morning, in the winter shade,
On the first of March, on the holiday,
I thought I saw you breathing”
re: Stagger Lee
According to The Annotated Grateful Dead Lyrics site:
[QUOTE=The Annotated Grateful Dead Lyrics]
There are a number of tunes entitled “Stagger Lee” which have been recorded over the years. They all derive from a series of tales and songs in African- American folklore.
[/QUOTE]
See the link for many more details.
That said, the words to the Dead’s version of the story are credited to Robert Hunter, one of the band’s standard lyricists. (I wonder if other versions feature Stagger Lee getting shot in the balls.) So, not a traditional song and not a cover.
One funny thing, to me: I have always thought that the song took place on December 24 and was surprised when **Misnomer **put it under December 25. The opening lyric I hear on every version of the Dead performing this song is: “1940 Xmas eve with a full moon over town.” The published lyrics say otherwise, so…
I put it under December 25 because your post said, “1940 Xmas evening with a full moon over town.” Christmas evening is not the same as Christmas Eve. Now that you’ve clarified, I’ve changed it. 
Here’s the updated list:
Jan 01 - “New Year’s Day” - U2
Jan 30 - “Bloody Sunday” - Cruachan
Feb 14 - “Valentine’s Day” - Enuff Z’Nuff
Mar 01 - “Casimir Pulaski Day” - Sufjan Stevens
Mar 04 - “Radar” - Morphine
Mar 21 - “Town With No Cheer” - Tom Waits
Apr 01 - “April Fools” - Rufus Wainwright
Apr 01 - “The Ballad of Thunder Road” - Robert Mitchum
Apr 04 - “Pride (In The Name of Love)” - U2
Apr 15 - “One More Summer” - The Rainmakers
Apr 26 - “April 29, 1992 (Miami)” - Sublime
Apr 41 - “Wet Dreams” - Kip Addotta
May 01 - “May 1, 1990” - Adrian Belew
May 01 - “First of May” - Bee Gees
May 05 - “Isis” - Bob Dylan
May 10 - “The Night They Drove Old Dixie Down” - The Band
May 21 - “Great King Rat” - Queen
May 23 - “Schticks of One Kind and Half a Dozen of the Other” - Allan Sherman
May 24 - “Bye Bye Pride” - Go-Betweens
Jun 03 - “Ode to Billie Joe” - Bobby Gentry
Jun 03 - “Desiree” - Neil Diamond
Jun 06 - “Convoy” - CW McCall
Jun 23 - “Spancil Hill” - Michael Considine
Jun 30 - “The Last Day of June, 1934” - Al Stewart
Jul 04 - “Saturday In The Park” - Chicago
Jul 04 - “Fourth of July” - Dave Alvin
Jul 04 - “The Cuckoo” - Donovan/Bob Dylan
Jul 04 - “The Subway Song” - Tom Paxton
Jul 04 - “4th of July, Sandy” - Bruce Springsteen
Jul 04 - “Born on the 4th of July” - Tom Paxton
Jul 07 - “The 7th Day of July, 1777” - King Diamond
Jul 12 - “The Sash” - Irish
Aug 01 - “I Live With It Every Day” - Barenaked Ladies
Aug 05 - “Who Killed Marilyn” - The Misfits
Aug 07 - “August 7, 4:15” - Jon Bon Jovi
Aug 08 - “August 8” - NOFX
Sep 03 - “Papa Was A Rolling Stone” - The Temptations
Sep 06 - “Mission Viejo” - Lifter Puller
Sep 21 - “September” - Earth, Wind and Fire
Sep 22 - “Crying” - M2M
Sep 23 - “The 23rd of September” - Vulgar Boatmen
Oct 01 - “Schticks of One Kind and Half a Dozen of the Other” - Allan Sherman
Oct 31 - “The New Style” - Beastie Boys
Nov 05 - “Remember” - John Lennon
Nov 14 - “Things In My Life” - Men Without Hats
Nov 22 - “November 22, 1963” - Broken Arrow Music
Nov 23 - “Hijack” - Jefferson Starship
Dec 01 - “Sweet Baby James” - James Taylor
Dec 07 - “Pearl Harbor” - Final War
Dec 24 - “White Christmas” - The Carpenters
Dec 24 - “Stagger Lee” - Grateful Dead
Dec 25 - “When Jesus Was Born” - Fairfield Four
Dec 25 - “Christmas, 1915” - Celtic Thunder
Dec 31 - “What Are You Doing New Year’s Eve” - Ella Fitzgerald
If you accept multiple entries for dates, I’m sure that there are many for 25 December. For instance that Christmas classic by the Pogues and Kirsty MacColl, Fairytale of New York: "And the boys of the NYPD choir’s still singing Galway Bay - And the bells were ringing out for christmas day. "
“The 33rd of August” - Micky Newbury.
The opening of the the original (Fast Records) ‘Circus Of Death’ by The Human League:
This is a song called The Circus Of Death.
It tells the true story of a circus we met.
The first two verses concern the actual arrival at Heathrow Airport of Commissioner Steve McGarrett.
The third emotionally describes a map showing the range of the circus.
The fourth and fifth were extracted from an article in the Guardian of March the 19th, 1962.
The last is a short wave radio message from the last man on Earth.
Likewise:
“Everyone would gather on the 24th of May
Sitting in the sand to watch the fireworks display”
– Rush, “Lakeside Park”
“Yesterday…all my troubles seemed so far away…”
That’ll fill in all the empty dates, no?
I’m a pretty big music fan and I thought this would be easier, after 10 minutes of thought this is all I have:
Drama Rama, “What Are We Gonna Do”
It’s April 21st and everybody knows today is Earth Day
Merry Christmas, Happy Birthday to whoever’s being born
14th of July - Bastille Day by Rush
Jethro Tull’s “Ring Out Solstice Bells” was a UK hit in December 1976. The winter solstice in that year fell on December 21st.
That may be stretching the OP’s definition a bit…
If you are willing to include french music from the jazz era, Edith Piaf’s “padam, padam, padam” mentions 14th of July (Bastille day). The line goes “Des “je t’aime” de quatorze-juillet”, which means "The “I love you"s on Bastille day” .
March 11 - “Poofter’s Froth Wyoming Plans Ahead,” Frank Zappa & Captain Beefheart
March 16 - “Jesus on the Radio” by Guster
A repeat, but the Grateful Dead’s “Jackstraw” includes “Leaving Texas, Fourth Day of July…”
(… and I have a nagging feeling there’s at least one Richard Thompson song with a date in it, but can’t put my finger on it. )
“16th of June, nine 0 five, doorbell rings
Man at the door says if I want to stay alive a bit longer
There’s a few things I need you to know”
Breathe–U2
Paul kelly - ‘How to Make Gravy’
21st December
Artist: Scraping Foetus Off The Wheel
Song: I’ll Meet You In Poland Baby
Lyric: “Today is the first of September”
The Song of the Seabees
And we promise that we remember
The “Seventh of December”
September by Earth, Wind and Fire:
Do you remember the
21st night of September?
Love was changing the minds of pretenders
While chasing the clouds away
Alice’s Restaurant specifically mentions Thanksgiving Day. But maybe that doesn’t count since it doesn’t mention the actual date, and it’s different from year to year.