Uniruledness of some moduli spaces of pointed spin curves

The moduli space $\mathcal{S}_{g, 2n}$ parametrizes pointed curves with spin structure. These are tuples $[C, p_1, \dots, p_{2n}, \eta]$ where $\eta \in \text{Pic}(C)$ such that $\eta^{\otimes 2} \cong \omega_C(-p_1 - \dots - p_{2n})$. We prove that $\mathcal{S}_{2, 4}$, $\mathcal{S}_{2, 6}$, $\mathcal{S}_{3, 2}$, $\mathcal{S}_{3, 4}$, $\mathcal{S}_{3, 6}$, $\mathcal{S}_{4, 2}$, $\mathcal{S}_{4, 4}$, $\mathcal{S}_{5, 2}$ and $\mathcal{S}_{5, 4}$ are uniruled.