Percolation is a statistical concept that describes the formation of an infinite cluster of connected particles or pathways. Lowering the percolation threshold is a critical issue to achieve light and low-cost conductive composites made of an insulating matrix loaded with conductive particles. This has interest for applications where charge dissipation and electrical conductivity are sought in films, coatings, paints, or composite materials. One route to decreasing the loading required for percolation is to use rod-like particles. Theoretical predictions indicate that this may also be achieved by altering the interaction potential between the particles. Although percolation may not always respond monotonically to interactions, the use of adhesive rods can be expected to be an ideal combination. By using a system made of carbon nanotubes in an aqueous surfactant solution, we find that very small attraction can markedly lower the percolation threshold. The strength of this effect can thereby have direct technological interest and explain the large variability of experimental results in the literature dealing with the electrical behavior of composites loaded with conducting rods.