Theorem 5.9 in de Jong's paper is pretty general ($S$ need not be local, only excellent integral of finite dimension) with $X$ proper over $S$. If $X$ is only of finite type over $S$ but separated, using a compactification $\overline{X}$ of $X$, you should get alterations $X' \to X$, $S' \to S$ with $S'\to S$ generically étale.
EDIT: change ''birational'' to ''generically finite''
If you want $S'\to S$ be étale, it is possible just by shrinking $S'$ to the étale locus. If you want the condition on the fibers, it should still be possible by shrinking $S'$ (working with proper $X$, the locus where the fiber is not generically finite is closed and projects to a closed subset in $S$, the same holds for fibers $X'\to S'$ which are not regular, at least if the residue fields of $S$ are perfect). But if you want $S'\to S$ be surjective, it is impossible in general: let $S$ be the spectrum of a DVR with perfect residue field, what you want would imply that the generic fiber has potentially good reduction. This is false in general if $X$ is proper (with integral fibers to make sense for "fiberwise alteration").