Surjective proper cover by ordinary varieties

Question

Let $X$ be smooth proper variety over a finite field $k$ of positive characteristic $p$. Assume that $X$ is not ordinary, then my question is if there exists a smooth, projective ordinary $Y$ with a surjective proper morphism $Y\to X$. A (naive) approach would be to mimick what we do in the resolution of singularity by blowing up $X$ along a subvariety. However, for instance by Prop 4.2.2 here, the blow-up along $Z\subset X$ is ordinary if and only if both $X,Z$ are. I had a hope that I could maybe understand the situation better for the case of abelian varieties, but must admit to being stuck. I'd appreciate any input, I'm not quite sure what to believe.