New correlations are presented for the truncated Gaussian (TG) thermal contact conductance model during first loading. The TG model is also incorporated into existing models for the hysteresis effect of thermal contact conductance and for the plastic contact pressure. The TG models, as well as the existing fully Gaussian models, are compared against new experimental data collected at very low contact pressures. Comparison between the models and the data shows that the fully Gaussian model underpredicts data at low contact pressures, as already extensively reported in the literature. The first-loading TG model predicts the experiments very well over the entire range of the contact pressures tested. The hysteresis effect model proved to be accurate only for contact pressure above 400 kPa, in general. The TG model requires a surface roughness parameter, the level of truncation of the probability density function of surface heights, which cannot be obtained accurately from ordinary surface profilometry. The most accurate and straightforward way to estimate this surface geometry parameter is from thermal tests. The levels of truncation of bead-blasted stainless steel 304 and Ni 200 are obtained and presented.