Restricted frequency-domain specifications (RFDSs) are often encountered in control system design. In this technical note, we investigate the problem of static output-feedback (SOF) controller synthesis subject to a class of RFDSs. Motivated by the generalized Kalman-Yakubovich-Popov lemma and the two-stage strategy for SOF stabilization, a necessary and sufficient condition for the existence of an SOF controller satisfying an RFDS is first derived, and is then used to construct a heuristic approach to computing the SOF gain. The heuristic approach includes two stages: first designing an initial full-information (FI) controller and then computing an SOF gain. Moreover, a heuristic method is proposed to optimize the initial FI controller. Numerical examples are presented to demonstrate the effectiveness of the proposed design method. The main contribution of the technical note is extending the two-stage idea to SOF controller synthesis subject to RFDSs. In addition, the underlying relationship between some typical two-stage approaches to SOF stabilization is revealed.