  We extend the construction of Calabi-Yau manifolds to hypersurfaces in
non-Fano toric varieties, requiring the use of certain Laurent defining
polynomials, and explore the phases of the corresponding gauged linear sigma
models. The associated non-reflexive and non-convex polytopes provide a
generalization of Batyrev's original work, allowing us to construct novel pairs
of mirror models. We showcase our proposal for this generalization by examining
Calabi-Yau hypersurfaces in Hirzebruch n-folds, focusing on n=3,4 sequences,
and outline the more general class of so-defined geometries.
