Let $X\subset\mathbb{P}^n$ be an irreducible variety, and let $Sing(X)$ be its singular locus. Let $Y$ be the blow-up of $\mathbb{P}^n$ along $Sing(X)$. Assume that we know that the strict transform of $X$ in $Y$ is smooth.
Now, let $W\subset\mathbb{P}^n$ be a smooth variety such that $Sing(X)\subset W\subset X$, and let $Z$ be the blow-up of $\mathbb{P}^n$ along $W$. Is it true that the strict transform of $X$ in $Z$ is smooth?