Unconscious but recovering Coulson is being kept hidden is a SHIELD hospital, and the Avengers all believe it when Fury tells them Coulson's dead. Tony, with Jarvis's help, thinks differently, and the Avengers sneak into the hospital and find Coulson. The hospital is not too far from NYC. They enter the hospital through a forgotten tunnel, the entrance of which is located in a swampy area.
I think part of this story has one or more scenes where the Avengers are sleeping in the hospital overnight, and Tony's having difficulty sleeping due to there being too many reminders of his time in Afghanistan (maybe it's being alone in the dark). Someone stays in the room with him, so he's not alone, and he's able to sleep.
This is NOT the story where Jasper and Melinda May figure it out, and drive in through the front gate + where Thor crushes two stones on Coulson to heal his wounds then Mjolnirs him out of a coma.
Story has been found. It's Awakenings, by Brooke_Lynne