Experimental and theoretical work has suggested that protein crystal nucleation can be affected by the separation of two metastable liquid phases with different local concentrations, or more specifically by critical density fluctuations. We measure the amplitude and correlation length of local concentration fluctuations by light scattering for supersaturated solutions of hen egg-white lysozyme (at pH 4.5 and at different NaCl concentrations, up to 7% w/v). By extrapolating the critical divergent behavior of concentration fluctuation amplitude versus temperature, we determine the spinodal line, that is the limit of stability. Cloud-point measurements are used to determine liquid-liquid coexistence, consistent with previous work. In the present work, which is an extensive study of off-critical fluctuations in supersaturated protein solution, we observe a nonclassical scaling divergent behavior of the correlation length of concentration fluctuations, thus suggesting that off-critical fluctuations may have a role in crystallization kinetics. To appropriately fit the spinodal data, an entropic term must be added to the van der Waals or to the adhesive hard-sphere model. We interpret this contribution as due to the salt-induced modulation of protein hydration.