Writing $\mathcal{L}=(\mathcal{A}+\mathcal{L})-\mathcal{A}$ for some sufficiently ample $\mathcal{A}\in\Pic(\mathcal{X})$ reduces us to the case where $\mathcal{L}$ is very ample. We then get a closed embedding $i:\mathcal{X}\hookrightarrow\mathbf{P}^{N}_{k}\times_{k}S$ over $S$ such that $\mathcal{L}$ coincides with the restriction of $\mathcal{O}(1)$. Let $\pi:\mathbf{P}^{N}_{k}\to\mathbf{P}^{N}_{k}$ be the morphism $[X_{0}:\dots:X_{N}]\mapsto[X_{0}^{m}:\dots:X_{N}^{m}]$, which satisfies $\pi^{*}\mathcal{O}(1)=\mathcal{O}(m)$. For each $\sigma\in\mathrm{PGL}(N+1,k)$ set $i_{\sigma}:=\sigma\circ i$ and consider $\mathcal{X}^{\prime}:=\mathcal{X}\times_{i_{\sigma}}\pi$ with the finite surjective morphism $\rho:\mathcal{X}^{\prime}\to\mathcal{X}$, so that $\rho^{*}\mathcal{L}=\mathcal{O}(m)|_{\mathcal{Y}}$ is divisible by $m$ in $\Pic(\mathcal{Y})$.

Applying Kleiman’s Bertini-type theorem (cf. [Har77, III.10.8]) to the smooth $k$-varieties $E_{J}=\bigcap_{j\in J}E_{j}$ for all subsets $J\subset I$ shows that we may choose $\sigma\in\mathrm{PGL}(N+1,k)$ such that each $\rho^{*}E_{i}$ is smooth over $k$ and $\sum_{i}\rho^{*}E_{i}$ has simple normal crossings. This implies in particular that $\mathcal{X}^{\prime}$ is an SNC model. ∎