I think this question is asking for something too strong; here's an example. Let $R = \mathbf{Z}_p$, and let $S_0$ be a finite flat $\mathbf{F}_p$-algebra that does not lift to a finite flat $R$-algebra; an explicit example can be found here. Since $R$ is local, the present question asks: does there exist a faithfully flat $R$-algebra $S$ lifting $S_0$? If there was such an $S$, then the $p$-adic completion $\widehat{S}$ of S would be a finite flat R-algebra lifting S_0 $S_0$ (the flatness is clear, and the finiteness comes from Nakayama, completeness, and finiteness modulo $p$), which cannot exist. Hence, there is no such $S$ either.
I think this question is asking for something too strong; here's an example. Let $R = \mathbf{Z}_p$, and let $S_0$ be a finite flat $\mathbf{F}_p$-algebra that does not lift to a finite flat $R$-algebra; an explicit example can be found here. Since $R$ is local, the present question asks: does there exist a faithfully flat $R$-algebra $S$ lifting $S_0$? If there was such an $S$, then the $p$-adic completion $\widehat{S}$ of S would be a finite flat R-algebra lifting S_0 (the flatness is clear, and the finiteness comes from Nakayama, completeness, and finiteness modulo $p$), which cannot exist. Hence, there is no such $S$ either.