Let $X$ be a reduced projective scheme of pure dimension 1 over the field $k$. Let $\pi: X \to \mathbb{P}_k^1$ be a finite, flat and surjective morphism onto the projective line.

What is the minimal $d \in \mathbb{Z}_{\geq 1}$ such that $H^1(X,\pi^*\mathcal{O}_{\mathbb{P}_k^1}(d)) = 0$?

I could already show that $d \geq -\chi(\mathcal{O}_X)/\deg(\pi)$ is necessary. But I was wondering whether $d \in O(-\chi(\mathcal{O}_X)/\deg(\pi))$ holds.

I am grateful for any kind of help or maybe even an example of such a situation where $d \notin O(-\chi(\mathcal{O}_X)/\deg(\pi))$ holds.

Let $X$ be a reduced projective scheme of pure dimension 1 over the field $k$. Let $\pi: X \to \mathbb{P}_k^1$ be a finite, flat and surjective morphism onto the projective line.

What is the minimal $d \in \mathbb{Z}_{\geq 1}$ such that $H^1(X,\pi^*\mathcal{O}_{\mathbb{P}_k^1}(d)) = 0$?

I could already show that $d \geq -\chi(\mathcal{O}_X)/\deg(\pi)$ is necessary. But I was wondering whether $d \in O(-\chi(\mathcal{O}_X)/\deg(\pi))$ holds.

I am grateful for any kind of help or maybe even an example of such a situation where $d \notin O(-\chi(\mathcal{O}_X)/\deg(\pi))$ holds.

Edit: If $X$ is integral, then $d$ is in the desired order: We then have $$ H^1(X,\pi^*\mathcal{O}_{\mathbb{P}_k^1}(d)) \cong H^0(X,\omega_X \otimes_{\mathcal{O}_X}\pi^*\mathcal{O}_{\mathbb{P}_k^1}(-d)) $$ which is zero if $\deg \pi^*\mathcal{O}_{\mathbb{P}_k^1}(d) > \deg \omega_X = -2\chi(\mathcal{O}_X)$. So what happens otherwise?