The stabilization of chiral magnetic spin structures in thin films is often attributed to the interfacial Dzyaloshinskii-Moriya interaction (DMI). Very recently, however, it has been reported that the chirality induced by the DMI can be affected by dipolar interactions. These dipolar fields tend to form Néel caps, which entails the formation of a clockwise chirality at the top of the film and a counterclockwise chirality at the bottom. Here we show that engineering an alternating DMI that changes signs across the film thickness, together with the tendency to form Néel caps, leads to an enhanced stability of chiral spin-structures. Micromagnetic simulations for skyrmions demonstrate that this can increase the effective DMI in a prototypical [Pt/Co/Ir] multilayer system by at least 0.6mJm-2. These gains are comparable to what has been achieved using additive DMI, but more flexible as we are not limited to a select set of material combinations. We also present experimental results: By measuring equilibrium domain widths, we quantify the effective DMI in [Pt/Co/Ir] multilayer systems typically used for skyrmion stabilization. Upon introducing an alternating DMI, we demonstrate changes in the effective DMI that agree with our simulations. Our results provide a route toward enhancing the stability of chiral spin structures that does not rely on enlarging the chiral interactions.