The following argument should also work when $X$ has klt singularities.
By the result of Birkar-Cascini-Hacon-M$^{\rm c}$Kernaㄙ, we can find a birational model $f:X'\rightarrow X$ extracting only one divisor $E$ with $f(E)=W$. Since $-E$ is relative ample over $X$, there is an exact sequence, $$f_*\mathcal{O}_{X'}=\mathcal{O}_X\rightarrow f_*\mathcal{O}_{E}\rightarrow R^1f_*\mathcal{O}_{X'}(-E)=0.$$ Since there is a factorization $$\mathcal{O}_X\rightarrow \mathcal{O}_W\rightarrow f_*\mathcal{O}_E,$$ the natural map $E\rightarrow W$ has connected fibers. On the other hand, result of Hacon-M$^{\rm c}$Kernan (On Shokurov's rational connectedness conjecture, Duke Math. J.) asserts that $E\rightarrow W$ must have rationally connected fibers.
Now there are two situations where one can find a birational section:
when $W$ is also rationally connected, or
when $\dim W=1$.
These follows from the result of Graber-Harris-Mazur-Starr. To get a resolution of $X$, one can simply replace $X'$ by a higher model.
However, if $W$ is not rationally connected or of $\dim W\geq2$, it seems to me the answer is related to weak approximation problem as handled in the last paper. As I remember, this is a nontrivial problem.