Ha et al. [1] have constructed a novel set of smoothness indicators by using the L1-norm measure and introducing a user-tunable parameter to propose a new WENO scheme, dubbed WENO-NS. Kim et al. [2] have improved WENO-NS with another user-tunable parameter to balance the contribution of substencils, leading to the WENO-P scheme. Recently, Rathan et al. [3] have modified WENO-P by devising a higher-order global smoothness indicator. The associated MWENO-P scheme aimed to fix the issue of WENO-P and WENO-NS that they cannot achieve the optimal convergence orders at the critical points where the first and second derivatives are zero. The above-said schemes outperform the existing many WENO schemes in some senses. However, their smoothness indicators violate the symmetry-preserving property preserved by those of many other well-established WENO schemes and rely on one or two user-tunable parameters. In the present study, a new set of smoothness indicators satisfying the symmetry-preserving property without using any user-tunable parameters is constructed. The resultant scheme is called the improved MWENO-P scheme and abbreviated as IMWENO-P. We provide detailed theoretical analysis and numerical examples to verify that the new scheme can attain optimal convergence orders even in the presence of second-order critical points. Euler equations are simulated to demonstrate the enhancements of the new scheme, such as the lower dissipation and better resolution.