We generalize the temporally encoded spectral shifting (TESS) analysis for measuring plasma wakefields using spectral interferometry to dissimilar probe pulses of arbitrary spectral profile and to measuring nonlinear wakefields. We demonstrate that the Gaussian approximation used up until now results in a substantial miscalculation of the wakefield amplitude, by a factor of up to two. A method to accurately measure higher amplitude quasilinear and nonlinear wakefields is suggested, using an extension to the TESS procedure, and we place some limits on its accuracy in these regimes. These extensions and improvements to the analysis demonstrate its potential for rapid and accurate on-shot diagnosis of plasma wakefields, even at low plasma densities.