In this work, we have outlined an approach to account for third order corrections to the linear absorption spectrum computed in the widely-used cumulant framework. The approach takes into account the effect of moderately non-Gaussian energy gap fluctuations without exhibiting unphysical divergences and regions of negative absorbance in the resulting spectra. The method promises to yield more accurate and robust linear spectra in model systems and chromophores embedded in complex condensed phase environments alike.