Ample vector bundles with zero loci having a bielliptic curve section

Let X be a smooth complex projective variety and let $Z \subset X$ be a smooth submanifold of dimension $\geq 2$, which is the zero locus of a section of an ample vector bundle $\mathcal{E}$ of rank dim $X$ - dim $Z \geq 2$ on $X$. Let $H$ be an ample line bundle on $X$ whose restriction $H_Z$ to $Z$ is very ample. Triplets $(X, \mathcal{E}, H)$ as above are studied and classified under the assumption that $Z$ is a projective manifold of high degree with respect to $H_Z$, dmitting a curve section which is a double cover of an elliptic curve.