I’ll second “Angel” by Sara McLachlan and add “El Paso” by Marty Robbins, “Bullet The Blue Sky” by U2, “Space Odyssey” by David Bowie (not sure why, but for some reason I want that song played at my funeral), “Lucky Man” by ELO, “Angie” by the Stones and “Cat’s In The Cradle” by Harry Chapin. Honorable mention to “Last Kiss” by J Frank Wilson.
Why not? Weather’s good here in the fall. 
As for me, there are a number of sad songs I can think of, but I don’t think that’s what the OP is looking for. Still, at least two “non-sad” songs always bring me down a bit:
“This Shirt,” by Mary Chapin Carpenter.
“I Like Dreaming,” by Kenny Nolan.
No idea why.
I’m not a Christian, or in any way religious, but Mary, by Patty Griffin just kills me.
Jesus said, “Mother, I couldn’t stay another day longer”…
*weeps
Nothing completely sad to the lyrics of Desperado, but it sounds so mournful. I know it used to be a favorite to play at biker funerals.
That’s likely because the result of their death was that they never “came to their senses”!
Badumcha…
Try ze veel, I’ll be hier all zee veek…tvip your vaitress…
Ooh, good one. I haven’t heard Sting’s version, but Eva Cassidy’s is hard to listen to because it makes me feel so emotional.
Except them 
“Walk on by (wait on the corner)” Leroy van Dyke
An oldie, anyone remember it?
Lord Tanamo - I’m in the Mood For Love
It’s upbeat but I find it melancholy for some reason.
Back2Good by Matchbox 20
Ten Years Gone by Led Zeppelin
and…Something’s Always Wrong by Toad the Wet Sprocket
Judy Collins singing Amazing Grace.
“Over the Rainbow” as sung by the late great Israel Kanakawiwo’ole is the most beautiful version I’ve ever heard. If you saw 50 First Dates, it was played at the end. I do know whay it makes me feel sad, though. Well, maybe not sad, but nostalgic for Hawaii. I thougt of it because unbelievably, a local radio station here in Bangkok played it this morning!
“Take Five” by the Dave Brubeck Quartet. I’m not really sure why.
How do you feel about hearing “The Yellow Rose of Texas”? 
Must … not … bitch … anymore … about … Texas … Must … not … bitch … anymore … about … Texas …