Hilbert squares of degeneracy loci

Let $S$ be the first degeneracy locus of a morphism of vector bundles corresponding to a general matrix of linear forms in $\mathbb{P}^s$. We prove that, under certain positivity conditions, its Hilbert square $\mathrm{Hilb}^2(S)$ is isomorphic to the zero locus of a global section of an irreducible homogeneous vector bundle on a product of Grassmannians. Our construction involves a naturally associated Fano variety, and an explicit description of the isomorphism.