The Permuted Kernel Problem (PKP) is a computational problem for linear codes over finite fields that has emerged as a promising hard problem for constructing post-quantum cryptographic schemes, with its main application found in the digital signature scheme PERK, submitted to the NIST standardization process for quantum-secure additional signatures. Upon reviewing the first version of PERK, NIST recommended further research on the concrete complexity of PKP. In this work, we follow this recommendation and investigate algorithmic improvements to the known methods for solving PKP. Specifically, we build upon the state-of-the-art work of Santini, Baldi, and Chiaraluce (IEEE Trans. Inf. Theory, 2024), and introduce a new algorithm that outperforms it over a wide range of parameters, yielding double-digit bit reductions in estimated complexity on representative instances. Nevertheless, our analysis shows that these improvements do not affect the parameter-set choices in PERK, thereby reinforcing confidence in its security.