We construct models for the level by level equivalence between strong compactness and supercompactness in which for κ the least supercompact cardinal and δ ≤ κ any cardinal which is either a strong cardinal or a measurable limit of strong cardinals, 2δ > δ+ and δ is < 2δ supercompact. In these models, the structure of the class of supercompact cardinals can be arbitrary, and the size of the power set of κ can essentially be made as large as desired. (...) This extends and generalizes [5, Theorem 2] and [4, Theorem 4]. We also sketch how our techniques can be used to establish a weak indestructibility result. (shrink)