We microscopically derive the spin diffusion equation in an s-wave superconductor in the vicinity of the superconducting transition temperature T-c. Applying the general relation between the relaxation function and the response function to the present spin diffusion problem, we examine how the spin relaxation time and the spin diffusion coefficient are renormalized in the superconducting state. The analysis reveals that, below T-c, both the spin relaxation time and the spin diffusion coefficient are increased, resulting in an enhancement of the spin diffusion length. The present result may provide an explanation for the recent observation of an enhanced spin pumping signal below T-c in a Py/Nb/Pt system that is free from the coherence peak effect.