We set up a MUSCL approach for the 2D DDFV-like scheme introduced in [Goudon, Llobell, Minjeaud, 2022] for the full Euler equations, discretized with unstructured meshes. It is a finite volume staggered discretization framework where numerical densities, energies and velocities are stored on different locations. For the MUSCL scheme, a combination of several approaches is needed for the reconstruction of the densities and the velocities, in order to reach the second order accuracy. This work focuses on the reconstruction of the mass density and internal energy. The improved accuracy is already sensitive on problems containing contact discontinuities. We provide a set of numerical simulations to illustrate the enhanced accuracy.