In a recent paper, we have studied the enumeration of Hamiltonian cycles (abbreviated HCs) on the grid cylinder graph Pm+1 x Cn, where m grows while n is fixed. In this sequel, we study a much harder problem of enumerating HCs on the same graph only this time letting n grow while m is fixed. We propose a characterization for non-contractible HCs which enables us to prove that their numbers hnc, m (n) satisfy a recurrence relation for every fixed m. From the computational data, we conjecture that the coefficient for the dominant positive characteristic root in the explicit formula for hnc,m (n) is 1.