During the past two decades there has been increased interest in the optical excitation of surface plasmon resonance (SPR) at a metal-dielectric interface. This is due in large part to its potential applications in such areas as medical diagnostics and pharmaceutical research. Also occurring during this time has been a growing recognition by the quantum physics community that weak value amplification (WVA) can serve as a valuable metrological research resource. Recently WVA has been used to amplify very small optical Goos-Hänchen (GH) shifts in glass and it has also been shown that SPR can greatly enhance optical GH shifts at the metal/air interface in Kretschmann-Raether (KR) devices. This paper demonstrates experimentally the WVA of an off-resonance GH shift in a KR device and explains why WVA of sufficiently SPR enhanced optical GH shifts cannot be achieved.