I am posting my comment as an answer. Quite generally, let $\nu:\widetilde{X}\to X$ denote the blowing up of an ideal sheaf $\mathcal{I}$ on a scheme $X$. Denote by $\mathcal{O}_{\widetilde{X}}(-\underline{E})$ the inverse image ideal sheaf $\nu^{-1}\mathcal{I}\cdot \mathcal{O}_{\widetilde{X}}$, and denote by $E\hookrightarrow \widetilde{X}$ the corresponding closed subscheme. The ideal sheaf $\mathcal{O}_{\widetilde{X}}(-\underline{E})$ is an invertible sheaf on $\widetilde{X}$. A morphism to $\widetilde{X}$ is $E$-flat if the pullback of the following injective sheaf homomorphism is still injective, $$\mathcal{O}_{\widetilde{X}}(-\underline{E}) \hookrightarrow \mathcal{O}_{\widetilde{X}}.$$
Consider a square-zero extension of Artinian local rings, $$0 \to M \hookrightarrow A'\twoheadrightarrow A \to 0.$$ Denote the residue field $A/\mathfrak{m} = A'/\mathfrak{m}'$ by $A_0$.
Let $Y'\to \text{Spec}\ A'$ be a flat, finitely presented morphism. Denote $\text{Spec}\ A \times_{\text{Spec}\ A'} Y'$ by $Y$, and denote $\text{Spec}\ A_0\times_{\text{Spec}\ A'} Y'$ by $Y_0$, and similarly for other schemes over $\text{Spec}\ A'$.
Assume now that $X$ is a scheme over $\text{Spec}\ A'$ (e.g., redefine $X$ to equals $\text{Spec}\ A\times_{\text{Spec}\ k} X$ if $X$ is a $k$-scheme and $A'$ is a local Artinian $k$-algebra). Let $$f':Y'\to X,$$ be a morphism of $A'$-schemes. Define $Z'\hookrightarrow Y'$ to be the closed subscheme defined by the inverse image ideal sheaf of $\mathcal{I}$. Denote by $\mathcal{T}$ the kernel of the induced morphism, $$M\otimes_{A_0} \mathcal{O}_{Z_0} \twoheadrightarrow M\cdot \mathcal{O}_{Z'}.$$
Proposition. For every $X$-morphism, $$e:Y\to \widetilde{X},$$ that is $E$-flat, there exists an $X$-morphism $e':Y'\to \widetilde{X}$ extending $e$ if and only if the closed subscheme $Z'$ of $Y'$ defined by the inverse image ideal sheaf of $\mathcal{I}$ is $A'$-flat, and in this case $e'$ is also $E$-flat. Moreover, this holds if and only if $\mathcal{T}$ is the zero sheaf.
Proof. This follows quickly from the universal property of the blowing up, cf. the answer to the following MathOverflow question: Which functor does the blowing up represent?. QED
There exists an initial $A_0$-module quotient, $$q:M \twoheadrightarrow M_e,$$ such that the following composition is the zero homomorphism, $$\mathcal{T} \hookrightarrow M\otimes_{A_0} \mathcal{O}_{Z_0} \twoheadrightarrow M_e\otimes_{A_0} \mathcal{O}_{Z_0}.$$ The induced pushout of $A'$, $$A'_e := (A'\oplus M_e)/\Delta(M) = A'/\text{Ker}(q),$$ is the initial quotient of $A'$ such that $e$ extends to an $X$-morphism on $\text{Spec}\ A'_e \times_{\text{Spec}\ A} Y'.$
For more on the "obstruction" to the extension of $e$ given by this element in $\text{Hom}(\mathcal{T},\mathcal{O}_{Z_0})\otimes_{A_0} M$, please confer Section 2 of the following.
MR2007396 (2004i:14002)
Olsson, Martin; Starr, Jason
Quot functors for Deligne-Mumford stacks.
Special issue in honor of Steven L. Kleiman.
Comm. Algebra 31 (2003), no. 8, 4069–4096.
https://math.berkeley.edu/~molsson/quot2a.pdf