Hybrid nanostructured materials comprising block copolymers, nanoparticles, and lithium salts have the potential to serve as electrolytes in non-flammable rechargeable lithium batteries. Here we show that the addition of functionalized nanoparticles, at an optimized concentration, into lamellar block copolymer electrolytes, results in an increase in ionic conductivity. This is due to the occurrence of a lamellar-to-bicontinuous phase transition, driven by the addition of nanoparticles. The magnitude of the increase in conductivity is consistent with a simple model that accounts for the morphology of the conducting channels. The conductivity of the optimized hybrid electrolyte is only 6% lower than that of an idealized nanostructured electrolyte with perfectly connected conducting pathways and no dead ends.