In this paper we calculate additional contributions to the part of the non-Gaussianity of the primordial curvature perturbation ζ that comes from the three-point correlator of the field perturbations. We estimate this contribution using the following models for the origin of ζ: single-component inflation, multicomponent chaotic inflation, a two-component 'hybrid' inflationary model and the curvaton scenario. In all of these models, the additional contributions to the primordial non-Gaussianity considered here are too small to ever be detected.