We will discuss cardinals that we may call superstrongly unfoldable cardinals. A cardinal kappa is superstrongly unfoldable if it is theta-superstrongly unfoldable for every ordinal theta, meaning that for subset A of kappa there is a nontrivial elementary embedding j:M–>N between transitive ZFC models having critical point kappa such that j(kappa)>theta and V_{j(kappa)} is a subset of N and the set A is an element of M. Superstrongly unfoldable cardinals lie in consistency strength between weakly compact cardinals and Ramsey cardinals, and they are relatively consistent with V=L. We will show the tighter consistency bounds of strongly unfoldable cardinals below and subtle cardinals above. This is joint work with Joel David Hamkins.