When solving inequality constrained optimization problems via Sequential Quadratic Programming (SQP), it is potentially advantageous to generate iterates that all satisfy the constraints: all quadratic programs encountered are then feasible and there is no need for a surrogate merit function. It has recently been shown that this is indeed possible, by means of a suitable perturbation of the original SQP iteration, without losing superlinear convergence. In this context, the well known Maratos effect is compounded by the possible infeasibility of the full step of one even close to a solution. These difficulties have been accommodated by making use of a suitable modification of a bending" technique proposed by Mayne and Polak, requiring evaluation of the constraints function at an auxiliary point at each iteration. In part I of this two-part paper, it was shown that, when feasibility of the successive iterates is not required, the Maratos effect can be avoided by combining Mayne and Polak's technique with a nonmonotone line search proposed by Grippo, Lampariello and Lucidi in the context of unconstrained optimization, in such a way that, asymptotically, function evaluations are no longer performed at auxiliary points. In this second part, we show that feasibility can be restored without resorting to additional constraint evaluations, by adaptively estimating a bound on the second derivatives of the active constraints. Extension to constrained minimax problems is briefly discussed.