Spin excitations are one of the top candidates for mediating electron pairing in unconventional superconductors. Their coupling to superconductivity is evident in a large number of systems, by the observation of an abrupt redistribution of magnetic spectral weight at the superconducting transition temperature, T(c), for energies comparable to the superconducting gap. Here we report inelastic neutron scattering measurements on Fe-based superconductors, Fe(1+y-x)(Ni/Cu)(x)Te(0.5)Se(0.5) that emphasize an additional signature. The overall shape of the low energy magnetic dispersion changes from two incommensurate vertical columns at T≫T(c) to a distinctly different U-shaped dispersion at low temperature. Importantly, this spectral reconstruction is apparent for temperatures up to ~3T(c). If the magnetic excitations are involved in the pairing mechanism, their surprising modification on the approach to T(c) demonstrates that strong interactions are involved.