Low Ambiguity Zone (LAZ) sequences play a pivotal role in modern integrated sensing and communication (ISAC) systems. Recently, Wang \textit{et al.} [arXiv:2501.11313] proposed a definition of locally perfect nonlinear functions (LPNFs) and constructed three classes of both periodic and aperiodic LAZ sequence sets with flexible parameters by applying such functions and interleaving techniques. Some of these LAZ sequence sets are asymptotically optimal with respect to the Ye-Zhou-Fan-Liu-Lei-Tang bounds under certain conditions. In this paper, we present constructions of three new classes of LPNFs with new parameters. Based on these LPNFs, we further propose a series of LAZ sequence sets that offer more flexible parameters. Furthermore, our results show that some of these classes are asymptotically optimal in both the periodic and aperiodic cases, respectively.