$S$-almost perfect commutative rings

Given a multiplicative subset $S$ in a commutative ring $R$, we consider $S$-weakly cotorsion and $S$-strongly flat $R$-modules, and show that all $R$-modules have $S$-strongly flat covers if and only if all flat $R$-modules are $S$-strongly flat. These equivalent conditions hold if and only if the localization $R_S$ is a perfect ring and, for every element $s\in S$, the quotient ring $R\mathrm{/}sR$ is a perfect ring, too. The multiplicative subset $S\subset R$ is allowed to contain zero-divisors.