In a previous paper, we have achieved the performance analysis of staggered Lagrange-remap schemes, a class of solvers widely used for Hydrodynamics applications. This paper is devoted to the rethinking and redesign of the Lagrange-remap process for achieving better performance using today's computing architectures. As an unintended outcome, the analysis has lead us to the discovery of a new family of solvers -- the so-called Lagrange-flux schemes -- that appear to be promising for the CFD community.