The heating of a solid wire embedded in a solid substrate (of lower Z material) with relativistic electrons generated by ultra-intense laser irradiation is considered. Previously, it has been noted that the initial angular distribution of the fast electrons is a highly important factor in the efficacy of the heating [Robinson et al., Phys. Plasmas 20, 122701 (2013)]. We show that, using 3D numerical simulations, the addition of an inverse conical taper at the front of wire can considerably improve the heating of the wire due to the reduction of angular spread of the fast electrons which is caused by transport through the inverse conical taper [Robinson et al., “Guiding of laser-generated fast electrons by exploiting the resistivity-gradients around a conical guide element,” Plasma Phys. Controlled Fusion (to be published)].