A system for cooling a wall (24) of a component having an outer surface with raised ribs (12) defining a structural pocket (10), including: an inner wall (26) within the structural pocket and separating the wall outer surface within the pocket into a first region (28) outside of the inner wall and a second region (40) enclosed by the inner wall; a plate (14) disposed atop the raised ribs and enclosing the structural pocket, the plate having a plate impingement hole (16) to direct cooling air onto an impingement cooled area (38) of the first region; a cap having a skirt (50) in contact with the inner wall, the cap having a cap impingement hole (20) configured to direct the cooling air onto an impingement cooled area (44) of the second region, and; a film cooling hole (22) formed through the wall in the second region.