CP-based Lagrangian relaxation (CP-LR) is an efficient optimization technique that combines cost-based filtering with Lagrangian relaxation in a constraint programming context. The state-of-the-art filtering algorithms for the WeightedCircuit constraint that encodes the traveling salesman problem (TSP) are based on this approach. In this paper, we propose an improved CP-LR approach that locally modifies the Lagrangian multipliers in order to increase the number of filtered values. We also introduce two new algorithms based on the latter to filter WeightedCircuit. The experimental results on TSP instances show that our algorithms allow significant gains on the resolution time and the size of the search space when compared to the state-of-the-art implementation.