We build on our previous work to derive more accurate analytical approximations for the phase-plane trajectories of the standard susceptible-infectious-removed (SIR) epidemic model, including host births and deaths. From our refined analysis, we obtain closed-form analytical expressions for the maximum and minimum prevalence following an initial outbreak. As in our previous work, our analysis involves matching asymptotic expansions across branch cuts of the Lambert W function, but we carry the approximations to higher asymptotic orders.