  We perform a state-of-the-art global study of the cosmological thermal
histories of a simple Yukawa model, and find higher perturbative orders to be
important for determining both the presence and strength of strong first-order
phase transitions. Using high-temperature effective field theory, we calculate
the free energy density of the model up to $\mathcal{O}(y^5T^4)$, where $y$ is
the Yukawa coupling and $T$ is the temperature. The locations of phase
transitions are found using the results of lattice Monte-Carlo simulations, and
the strength of first-order transitions are evaluated within perturbation
theory, to 3-loop order. This is the first global study of any model at this
order. Compared to a vanilla 1-loop analysis, accurate to $\mathcal{O}(y^2
T^4)$, reaching such accuracy enables on average a five-fold reduction in the
relative error in the predicted critical temperature $T_\text{c}$, and an
additional $\sim50\%$ strong first-order transitions with latent heat
$L/T_\text{c}^4 > 0.1$ to be identified in our scan.
