This paper considers the problem of downlink localization of user equipment devices (UEs) that are either in the near-field (NF) or in the far-field (FF) of the array of the serving base station (BS). We propose a dual signaling scheme, which can be implemented at the BS, for localizing such UEs. The first scheme assumes FF, while the other assumes NF conditions. Both schemes comprise a beam-sweeping technique, employed by the BS, and a localization algorithm, employed by the UEs. The FF-based scheme enables beam-steering with a low signaling overhead, which is utilized for the proposed localization algorithm, while the NF-based scheme operates with a higher complexity. Specifically, our proposed localization scheme takes advantage of the relaxed structure of the FF, which yields low computational complexity, but is not suitable for operating in the NF. Since the compatibility and the performance of the FF- based scheme depends on the BS-to-UE distance, we study the limitations of FF-based procedure, explore the trade-off in terms of performance and resource requirements for the two schemes, and propose a triggering condition for operating the component schemes of the dual scheme. Also, we study the performance of an iterative localization algorithm that takes into account the accuracy-complexity trade-off and adapts to the actual position of the UE. We find that the conventional Fraunhofer distance is not sufficient for adapting localization algorithms in the mixed NF and FF environment.