In paper I polynomial interpolating formulae of order 3 and 5 have been proposed and tested for transforming a non--linear differential Hamiltonian system into a map without having to integrate whole orbits as in the well known Poincar\'e return map technique. The precision of the computations increases drastically with the order of the polynomial fit which requires an extended amount of local information, i.e. information about neighbouring points. The first part of the paper deals with another type of interpolation where the information, within the same accuracy, refers only to the nearest neighbours but takes into account gradient information. The results are in very good agreement with those obtained using an order 3 symmetrical interpolation formula well inside the phase space. Moreover the new method is more effective at the border of the phase space when compared with asymmetrical interpolation. The second part of the paper deals with higher dimensional mappings, i.e. mappings for hamiltonian systems with 3 degrees of freedom.