  We demonstrate analytically and numerically that "optimized partial dressing"
(OPD) thermal mass resummation, which uses gap equation solutions inserted into
the tadpole, efficiently tames finite-temperature perturbation theory
calculations of the effective thermal potential, without necessitating use of
the high-temperature approximation. An analytical estimate of the scale
dependence for OPD resummation, standard Parwani resummation (Daisy
resummation), and dimensional reduction shows that OPD has similar scale
dependence to dimensional reduction, greatly improving Parwani resummation. We
also elucidate how to construct and solve the gap equation for realistic
numerical calculations, and demonstrate OPD's improved accuracy for a toy
scalar model. OPD's improved accuracy is most physically significant when the
high-temperature approximation breaks down, rendering dimensional reduction
unusable and Parwani resummation highly inaccurate, with the latter
underestimating the maximal gravitational wave amplitude for the model by 2
orders of magnitude compared to OPD. Our work highlights the need to bring
theoretical uncertainties under control even when analyzing broad features of a
model. Given the simplicity of the OPD compared to two-loop dimensional
reduction, as well as the ease with which this scheme handles departures from
the high-temperature expansion, we argue this scheme has great potential in
analyzing the parameter space of realistic beyond the Standard Model models.
