The Upwind Source at Interface method for hyperbolic conservation laws with source term introduced in [B. Perthame, C. Simeoni, Convergence of the Upwind Interface Source method for hyperbolic conservation laws, Hyperbolic Problems: Theory, Numerics, Applications (T. Hou and E. Tadmor, Eds.), Springer, 2003] is essentially first order accurate. Under appropriate hypotheses of consistency on the finite volume discretization of the source term, we prove $L^p$-error estimates, $1\le p < +\infty$, in the case of a uniform spatial mesh, for which an optimal result can be obtained. We thus conclude that the same convergence rates hold as for the corresponding homogeneous problem. To improve the numerical accuracy, we develop two different approaches of dealing with the source term and we discuss the question of deriving second order error estimates. Numerical evidence shows that those techniques produce high resolution schemes compatible with the Upwind Source at Interface method.