In this work, we present an all Mach number staggered pressure correction scheme with MUSCL-like convection fluxes for the compressible Euler and Navier-Stokes equations. This solver is to be used by the Institute of Radioprotection and Nuclear Safety (IRSN) for deflagrations simulations. It has been implemented in the IRSN open-source code CALIF3S. In a first step, a tentative velocity is computed using the momentum balance equation, where the beginning-of-step value of the pressure is used. Then, a correction equation is introduced so that the end-of-step velocity and pressure satisfy a discrete momentum balance equation. This correction equation is coupled with the mass and internal energy balances and the resulting system is solved. Because the internal energy balance equation is used instead of the total energy equation, corrective terms are added to ensure consistency. With an implicit-in-time first order upwinding of the numerical fluxes with respect to the material velocity only, the scheme has been proven to have a discrete solution, and to enjoy the following properties: to preserve the positivity of the density and of the internal energy, and to keep the pressure and velocity constant accross contact discontinuities [Grapsas16] . In addition the scheme is asymptotic preserving is the low Mach number limit, in the sense that, with a given discretization, the computed solution is observed to converge when the Mach number tends to zero to the solution of a stable scheme for the asymptotic problem. We build here a less diffusive variant of the scheme, by switching, for the convection fluxes in the mass and energy balances, from implicit-in-time upwind numerical fluxes to explicit-in-time MUSCL-like numerical fluxes [Piar13, Gastaldo18]. We show that under a CFL-like condition that depends only on the material velocity, the properties stated above for the former upwind scheme are preserved. We also observe by numerical experiments that the scheme is still able to cope with low Mach number viscous and inviscid flows. Gains in accuracy are quantified using a manufactured analytical solution.