The Stacks project

Proof. This is an immediate corollary of Lemma 109.4.2. Namely, unwinding the definitions, this lemma gives there is a surjective smooth morphism $S' \to S$ such that $X' = X \times _ S S'$ comes endowed with an invertible $\mathcal{O}_{X'}$-module $\mathcal{L}'$ which is ample on $X'/S'$. Then we can refine the smooth covering $\{ S' \to S\}$ by an étale covering $\{ S_ i \to S\}$, see More on Morphisms, Lemma 37.38.7. After replacing $S_ i$ by a suitable open covering we may assume $X_ i \to S_ i$ is H-projective, see Morphisms, Lemmas 29.43.6 and 29.43.4 (this is also discussed in detail in More on Morphisms, Section 37.50). $\square$