Assume we are working over $\mathbb{C}$, and we have a projective morphism with connected fibers $f: X \rightarrow Z$ whose geometric generic fiber $X_\overline{\eta}$ is isomorphic to a Hirzebruch surface $\mathbb{F}_n$.

Thus, $X_\overline{\eta}$ admits a morphism to $\mathbb{P}^1$, and this is defined over some finite extension of $K(Z)$. So, we know that, up to a generically finite base change and birational modification of the main component of the fiber product, we get a morphism $f': X' \rightarrow Z'$ that factors as $g': X' \rightarrow Y'$ and $h': Y' \rightarrow Z'$, where $g'$ and $h'$ are both generically $\mathbb{P}^1$-bundles.

My question is the following. Given the setup above, are there cases when we know that the finite base extension is not needed? My naive hopes rely on two facts. First, the morphism $\mathbb{F}_n \rightarrow \mathbb{P}^1$ is defined over $\mathrm{Spec}(\mathbb{Z})$, and so we can base change it to $K(Z)$. Second, if the base $Z$ is a curve, by the theorem of Graber-Harris-Starr we know that $X_\eta$ has a $K(Z)$-point. For instance, is it reasonable to get something in the direction I want if the base is a curve?