In this paper, we present a fully local second-order upwind scheme, applicable on generic meshes. This is done by hybridisation, which is achieved by introducing unknowns on each edge of the mesh. By doing so, fluxes only depend on values associated to a single cell, and thus, this scheme can easily be applied even on cells near the boundary of the domain. Another advantage of hybridised schemes is that static condensation can be employed, leading to a very efficient implementation. A convergence analysis, which also covers a flux-limited TVD variant of the scheme, is then presented. Numerical results are also given in order to compare this with a hybridised first-order upwind scheme and a classical cell-centered second-order upwind type scheme.