In this proof we will use that flat, finite presentation, finite locally free are properties that are preserved under base change and composition. We will also use that a finite locally free morphism is both open and closed. You can find these facts as Morphisms, Lemmas \ref{morphisms-lemma-base-change-flat}, \ref{morphisms-lemma-base-change-finite-presentation}, \ref{morphisms-lemma-base-change-finite-locally-free}, \ref{morphisms-lemma-composition-flat}, \ref{morphisms-lemma-composition-finite-presentation}, \ref{morphisms-lemma-composition-finite-locally-free}, \ref{morphisms-lemma-fppf-open}, and \ref{morphisms-lemma-finite-proper}. \medskip\noindent Note that $X_Z \to Z$ is flat morphism of finite presentation which has a section $s$ coming from $\sigma$. Let $X_Z^0$ denote the subset of $X_Z$ defined in Situation \ref{situation-connected-along-section}. By Lemma \ref{lemma-connected-along-section-open} it is an open subset of $X_Z$. \medskip\noindent The pullback $X_{Z \times_Y Z}$ of $X$ to $Z \times_Y Z$ comes equipped with two sections $s_0, s_1$, namely the base changes of $s$ by $\text{pr}_0, \text{pr}_1 : Z \times_Y Z \to Z$. The construction of Situation \ref{situation-connected-along-section} gives two subsets $(X_{Z \times_Y Z})_{s_0}^0$ and $(X_{Z \times_Y Z})_{s_1}^0$. By Lemma \ref{lemma-base-change-connected-along-section} these are the inverse images of $X_Z^0$ under the morphisms $1_X \times \text{pr}_0, 1_X \times \text{pr}_1 : X_{Z \times_Y Z} \to X_Z$. In particular these subsets are open. \medskip\noindent Let $(Z \times_Y Z)_y = \{z_1, \ldots, z_n\}$. As $X_y$ is geometrically connected, we see that the fibres of $(X_{Z \times_Y Z})_{s_0}^0$ and $(X_{Z \times_Y Z})_{s_1}^0$ over each $z_i$ agree (being equal to the whole fibre). Another way to say this is that $$ s_0(z_i) \in (X_{Z \times_Y Z})_{s_1}^0 \quad\text{and}\quad s_1(z_i) \in (X_{Z \times_Y Z})_{s_0}^0. $$ Since the sets $(X_{Z \times_Y Z})_{s_0}^0$ and $(X_{Z \times_Y Z})_{s_1}^0$ are open in $X_{Z \times_Y Z}$ there exists an open neighbourhood $W \subset Z \times_Y Z$ of $(Z \times_Y Z)_y$ such that $$ s_0(W) \subset (X_{Z \times_Y Z})_{s_1}^0 \quad\text{and}\quad s_1(W) \subset (X_{Z \times_Y Z})_{s_0}^0. $$ Then it follows directly from the construction in Situation \ref{situation-connected-along-section} that $$ p^{-1}(W) \cap (X_{Z \times_Y Z})_{s_0}^0 = p^{-1}(W) \cap (X_{Z \times_Y Z})_{s_1}^0 $$ where $p : X_{Z \times_Y Z} \to Z \times_W Z$ is the projection. Because $Z \times_Y Z \to Y$ is finite locally free, hence open and closed, there exists an affine open neighbourhood $V \subset Y$ of $y$ such that $q^{-1}(V) \subset W$, where $q : Z \times_Y Z \to Y$ is the structure morphism. To prove the lemma we may replace $Y$ by $V$. After we do this we see that $X_Z^0 \subset Y_Z$ is an open such that $$ (1_X \times \text{pr}_0)^{-1}(X_Z^0) = (1_X \times \text{pr}_1)^{-1}(X_Z^0). $$ This means that the image $X^0 \subset X$ of $X_Z^0$ is an open such that $(X_Z \to X)^{-1}(X^0) = X_Z^0$, see Descent, Lemma \ref{descent-lemma-open-fpqc-covering}. Finally, $X^0$ is quasi-compact because $X_Z^0$ is quasi-compact by Lemma \ref{lemma-connected-along-section-locally-constructible} (use that at this point $Y$ is affine, hence $X$ is quasi-compact and quasi-separated, hence locally constructible is the same as constructible and in particular quasi-compact; details omitted). In this way we see that $X^0$ has all the desired properties.