Multicarrier radar waveforms have many desirable properties for many radar applications. The well known multifrequency complementary phase-coded (MCPC) radar waveforms based on the consecutive ordered cyclic shift (COCS) design can achieve a low peak-to-mean envelope power ratio (PMEPR). In this paper, we emphasize that the low PMEPR in COCS design is achieved at a cost of degraded ambiguity function (AF) with high sidelobes. An efficient PMEPR reduction strategy should lower PMEPR while maintaining the desired AF. To address this, we propose a method for lowering the PMEPR using a constant envelope (CE) modulation and filtering operations. It is demonstrated that the proposed method is able to provide lower PMEPR than the COCS design without degrading the AF properties.