The pulse solution of the spatially discrete excitable FitzHugh-Nagumo (FHN) system is approximately constructed using matched asymptotic expansions in the limit of large time scale separation (as measured by a small dimensionless parameter epsilon). The pulse profile typically consists of slowly varying regions of the excitatory variable separated by sharp wave fronts. In the FHN system, the velocity of a pulse is decided by the interaction between its leading and trailing fronts, but the leading order approximation gives only a fair result when compared with direct numerical solutions. A higher order approximation to the wave fronts comprising the FHN pulse is found. Our approximation provides an c-dependent pulse velocity that approximates much better the velocity obtained from numerical solutions. As a result, the reconstruction of the FHN pulse using the improved wave fronts is much closer to the numerically obtained pulse.