Trent Reznor of Nine Inch Nails' song Hurt, as performed by Johnny Cash:

https://youtu.be/8AHCfZTRGiI

I think in order to get to the true depths of musical pain, you often need two people. One who has the creativity and expression to write it well, and another whose soul has been crushed by their own guilt enough to perform it well. These two people cannot be the same, because the soul crushing process kills your creativity at its most extreme levels. But that extreme level is where you can perform mournful songs the best.

So, it takes two people.

have a nice life (all of deathconsciousness but specifically who would leave their son out in the sun)

a silver mt. zion (all of he has left us alone...)

amenra (a solitary reign)

planning for burial (all of below the house)

lingua ignota (anything)

uboa (anything)

ex:re (all of S/T)

mount eerie (all of a crow looked at me)

panchiko (cut)

sufjan stevens (should have known better)

fugazi (I'm so tired)

nine inch nails (hurt)

midwife (all of like author, like daughter)

psychonaut 4 (all of dipsomania)

the whole genre of dsbm comes to mind

Casimir Pulaski Day by Sufjan Stevens. Turns out childhood cancer makes for a beautifully gut-wrenching song.

Can't believe no one has mentioned It's A Mad World yet:

https://youtu.be/4N3N1MlvVc4?si=-7kTHbNDPsNBHFuh

Also

Belly - Stay

Regina Spektor - Field Below

Tori Amos - Marianne, Twinkle, I'm Not In Love, Strange

Seether - Fine Again

PJ Harvey - To Bring You My Love

The album Hospice by the Antlers.

I'll just steal from wikipedia: Set in New York City's Memorial Sloan Kettering Cancer Center, which the second track is named after, Hospice tells the story of a relationship between a hospice worker and a female patient suffering from terminal bone cancer, their ensuing romance, and their slow downward spiral as a result of the woman's traumas, fears, and disease. The story of her deterioration also serves as a metaphor for an abusive relationship. Frontman Peter Silberman has been reluctant to divulge explicit details regarding the meaning of the record, and the extent to which it is autobiographical.

Love it.

I have been preparing my entire life for this moment!

Daughter - Shallows

Placebo - Follow The Cops Back Home

Lanterns on the Lake - Another Tale From Another English Town

Amanda Palmer - Trout Heart Replica

Joni Mitchell - The Magdelene Laundries (cw: abuse - if you're not looking for that sad try Come in From The Cold)

Secret Machines - 1000 Seconds

Silversun Pickups - Circadian Rhythm

Mogwai - Killing All The Flies

Porcupine Tree - Heart Attack in a Layby

Death Cab For Cutie - What Sarah Said

Iron & Wine - Passing Afternoon

Jimmy Eat World - Hear You Me

Tusks - Dissolve

The Flaming Lips - Mr Ambulance Driver

Tori Amos - Little Earthquakes

Manchester Orchestra - I Know How To Speak

The Midnight - Brooklyn (not to be confused with "Brooklyn. Friday. Love" which is the exact opposite end of the emotional spectrum)

Blue Valentines - Tom Waits

107 Steps - Björk (this one is best after watching the movie "Dancer in the Dark")

Gone Away - My Brightest Diamond

Gloomy Sunday - Billie Holiday Or Gloomy Sunday - Sinéad O'Connor No comparison vocally, but if you like Sinéad the second has a lot of emotion.

A Night Like This - The Cure

Traveling Light - Leonard Cohen

Heard the Friends theme song in minor?

I'd never really listened to the song before, but was struck with how melancholic the lyrics are.

Leviathan - At the door of the tenth sub-level of suicide

Here, try some doom black metal

Dido's Lament as performed by Simone Kermes from Henry Purcell's Dido and Aenaes

Clann Zu - One Bedroom Apartment

Death - Voice of the Soul

Austere - Just for a Moment

Eikenskaden - Then I'll Erase Myself Forever

Warning - Watching From a Distance

Brahams' Ein Deutsches Requiem - First Movement

Pearl Jam - Last Kiss

Dawn Penn - You Don't Love Me

The whole album, Asking for Flowers by Kathleen Edwards, but especially the song Alicia Ross, after you read the story behind it. That's weapons-grade sadness.

Bryan John Appleby - Honey Jars

Dance with the Devil by Immortal Technique. A song about an up-and-coming gangster being tricked into raping and killing his own mother. It's the most brutal, soul-crushing song I've ever heard.

The Cure - The Same Deep Water as You

Or just listen to Disintegration all the way through

The entire album The Final Cut by Pink Floyd

Yesterday When I Was Young (Roy Clark version, Glenn Campbell did this one too)

Tom Waits - Soldier's Things or Christmas Card from a Hooker in Minneapolis, or just throw on any Waits album and you'll find something unbearably sad among the bangers. Same with Nick Cave

If you don' t mind not understanding the lyrics, try the album Kauan by the finnish band Tenhi.

While were at finnish bands: some Nightwish songs are really great sad music: Away, Sleeping Sun and Deep Silent Complete come to my mind.

Also have a listen to Songs by John Dowland. Baroque but magnificentky sad if well sung.

Rehab - It Don't Matter

Alter Bridge - Blackbird

My favorite genre! I see you've already got Elliott Smith, so in the vein of hauntingly beautiful:

Greg Laswell - Your Ghost (A lot of his music is like this)
The National - I need my girl
Sun Kil Moon - Carry me Ohio

Other favorites
Local Natives - Heavy Feet
Boy & Bear - Wicked Game
Hurt - Alone with the Sea

Similar feel, but a little happier
Dope Lemon - Marinade

You should look into Eels. "Electro shock blues" is so sad and it starts with a down note on "Elizabeth on the bathroom floor".

"Sometimes It Snows in April" by Prince

Fleetwood Mac, Man of the World. “But I just wish that I'd never been born”. That one line alone does it for me.

lush - when I die

Give the album A Crow Looked at Me by Mount Eerie a listen if you haven't heard it yet.

Alice in Chains - Down in a Hole