It seems that you want a position where light can get emitted and then return to where it started and then continue on forever.
Such places do exist; what you're looking for is a circular null orbit. Uh, "null" means having no change in proper time, that is, the sort of path (or orbit or geodesic) that is suitable for light. The radius for this orbit is $3r_s/2$ where $r_s$ is the Schwarzschild radius. See wikipedia, Schwarzschild metric / orbital motion.
The spacecraft emitting the light would have to use thrusters (or something else, maybe a rope or what have you) to keep itself stationary, but the value is outside the event horizon at the Schwarzschild radius (where $r_s=1$), so this is possible.