The Goos-Hänchen (GH) shift describes a phenomenon in which a specularly reflected beam is translated along the reflecting surface such that the incident and reflected rays no longer intersect at the surface. Using a neutron spin-echo technique and a specially designed magnetic multilayer mirror, we have measured the relative phase between the reflected up and down neutron spin states in total reflection. The relative GH shift calculated from this phase shows a strong resonant enhancement at a particular incident neutron wave vector, which is due to a waveguiding effect in one of the magnetic layers. Calculations based on the observed phase difference between the neutron states indicate a propagation distance along the waveguide layer of 0.65 mm for the spin-down state, which we identify with the magnitude of the giant GH shift. The existence of a physical GH shift is confirmed by the observation of neutron absorption in the waveguide layer. We propose ways in which our experimental method may be exploited for neutron quantum-enhanced sensing of thin magnetic layers.