The Polynomial Modular Number Systems (PMNS) aim to represent elements of fields or rings of large characteristics using polynomials satisfying bounds on some parameters (degree, absolute values of the coefficients). Those PMNS, while some conditions are fulfilled, using convenient parameters and implementation features, allow some speed-ups in cryptographic computations. Recent works propose better speed-ups by improving the parameter generation of the system, and by using multiprecision polynomial coefficents i.e. coefficients stored using several machine words. In this work, we present a new improvement on the PMNS, applying to the internal reduction an approach similar to the truncated Montgomery reduction technique presented by Didier et al. In the context of software implementations using AVX512 instruction set extension, and modulo size up to 8192 bits, this new approach allows speed-ups up to 15% in modular multiplication computation, in comparison with conventional approaches.