The reappearance of order from chaos in two-degrees-of-freedom non-integrable Hamiltonians is shown not to be restricted to specially coupled nonlinear systems which effectively decouple on increasing the energy. The numerical demonstration is with a class of pair potentials constructed from the sum of three exponential terms. The coupling in these systems remains strong for all energies. These necessary physical condition on the bounded total potential VT for the reappearance of order is that in the limit of both small and large energy VT tends to an integrable limit. As a consequence, at least two order-to-chaos transition energies exist. A possible necessary condition on VT is given which may decide whether chaos is small scale or global. It is conjectured that there may be a fine structure to the degree of chaos vs E curves.