The hospital where Sayid and Nadia go to see Omer, and where Jack is briefly seen passing in the hallway (“Sundown,” Ep. 6×06), was Hawaii Medical Center East, formerly St. Francis Medical Center (2226 Liliha Street).
Note that the hospital later used for most of Season 6 was the Rehabilition Hospital of the Pacific in Nuuanu.
Map Coordinates: 21.329131, -157.85233
Google Street View (Entrance)
Episode image copyright ABC Studios, courtesy Lost-Media.com.