The initial stages of liquid-liquid phase separation in an off-critical binary polymer blend were studied by time-resolved small-angle neutron scattering (SANS). Phase separation is triggered by quenching homogeneous blends from the single-phase to the two-phase region of the phase diagram. Our objective was to determine the size of the critical nucleus, R-c, over a wide range of quench depths, kappa. We present results obtained from two-step quenches. In the first step, the blend is subjected to a deep quench, kappa', for a brief period of time. This is followed by a reduction in quench depth to kappa. One can view the first quench as a perturbation. We demonstrate that under certain conditions the only effect of the perturbation is to hasten nucleation kinetics at kappa. This allows the determination of R-c at low quench depths near the binodal, where nucleation barriers preclude the formation of viable nuclei after single-step quenches on experimental time scales. Our experiments at large kappa provide no evidence for a change in phase separation mechanism upon crossing the classical spinodal.