  We generalize the functorial quasi-isomorphism in \cite{Davis2011} from
overconvergent Witt de-Rham cohomology to rigid cohomology on smooth varieties
over a finite field $k$, dropping the quasi-projectiveness condition. We do so
by constructing an \et ale hypercover for any smooth scheme $X$, refined at
each level to be a disjoint union of open standard smooth subschemes of $X$. We
then find, for large $N$, an $N$-truncated closed embedding into a simplicial
smooth scheme over $W(k)$, which allows us to use the results of \textit{loc.
cit} at the simplicial level, and use cohomological descent to prove the
comparison.
