We prove that the moduli space $\overline{\mathbf{M}}_{0,0}(\mathcal{N},2)$ of stable maps of degree $2$ to the moduli space $\mathcal{N}$ of rank $2$ stable bundles of fixed odd determinant $\mathcal{O}_X(-x)$ over a smooth projective curve $X$ of genus $g\ge 3$ has two irreducible components that intersect transversely. One of them is Kirwan's partial desingularization $\widetilde{\mathcal{M}}_X$ of the moduli space $\mathcal{M}_X$ of rank $2$ semistable bundles with determinant isomorphic to $\mathcal{O}_X(y-x)$ for some $y\in X$ . The other component is the partial desingularization of the geometric invariant theory (GIT) quotient $\mathbb{P}\mathrm{Hom} ({\rm sl}(2)^\vee, \mathcal{W})//{\rm PGL}(2)$ for a vector bundle $\mathcal{W}=R^1{\pi}_*\mathcal{L}^{-2}(-x)$ of rank $g$ over the Jacobian of $X$ . We also show that the Hilbert scheme $\mathbf{H}$ , the Chow scheme $\mathbf{C}$ of conics in $\mathcal{N}$ , and $\overline{\mathbf{M}}_{0,0}(\mathcal{N},2)$ are related by explicit contractions