We describe improvements to the Kleinman-disallowed hyper-Rayleigh scattering (HRS) measurement technique, which can be used to measure all of the rotationally invariant figures of merit for the first molecular nonlinear optical hyperpolarizability tensor βijk. As multiphoton excited fluorescence continues to be an accuracy-limiting factor in any HRS experiment, we have implemented time-resolved single photon counting in HRS using a Ti:sapphire laser. We show, however, that spectral analysis is necessary to ensure that the fluorescence and second harmonic are emitted on different time scales in order to allow temporal separation and thus improved accuracy. We also demonstrate how an electrically controlled liquid-crystal phase retarder can be used to vary the polarization state of the probe light in order to determine the second-order nonlinear response tensor. It was found that results agree with those obtained using a rotating quarter-wave plate to control the polarization state and demonstrate improved precision and reproducibility.