Let $X$ be a smooth scheme over $\mathbb{C}$. Let us consider first-order deformations of $X$ over $S:=\operatorname{Spec}\mathbb{C}[t]/t^2$ i.e. flat surjective morphisms $\pi\colon \widetilde{X} \rightarrow S$, such that $\widetilde{X} \times_S \operatorname{Spec}(\mathbb{C}) \simeq X$. I do understand that to any such a morphism we may associate a short exact sequence $0 \rightarrow T_X \rightarrow T_{\widetilde{X}}|_X \rightarrow \pi^*(T_S)|_X \rightarrow 0$ that produces an element $\xi \in H^1(X,T_X)$ called Kodaira-Spencer class of the deformation $\pi$.
The question is the following: how to prove that the map $(\widetilde{X},\pi) \mapsto \xi$ is a bijection.
I do understand the complex-analytic proof of this statement (for example from Claire Voisen's book). It is very natural but it uses the fact that there exists a covering of $\widetilde{X}$ by open subsets $\widetilde{U}_i$ s.t. the deformations $\pi|_{\widetilde{U}_i}\colon \widetilde{U}_i \rightarrow S$ are trivial. The existence of such covering is not obvious for me in the algebraic context.