Provided are systems and methods incorporating design features that may be beneficial over current available well safety valves, such as a sleeved receptacle for a piston, a constrained piston rod, and a trapped flapper valve. A safety valve may comprise: an outer housing having a central bore extending axially through the outer housing, wherein the outer housing comprises an outer wall having a receptacle formed in the outer wall; a sleeve disposed in the receptacle; a piston disposed in the sleeve, wherein the piston is actuated to travel in the sleeve; a spring positioned to apply a biasing force to the piston; and a valve member disposed in the outer housing, wherein the valve member is operable to selectively restrict flow through the safety valve in response to movement of the piston.