Polymer electrolytes constitute an attractive alternative to current liquid electrolytes used in Li-ion batteries. Unfortunately, the lithium ion conductivities of the state-of-the-art polymer electrolytes are few orders of magnitude lower than those of liquid electrolytes at room temperature. In this work, we focus on poly(ethylene oxide) (PEO), which has shown the highest lithium ion conductivity in polymer electrolytes so far. At high salt concentrations, the lithium conductivity of a PEO electrolyte is strongly reduced due to the formation of ionic aggregates. Using molecular dynamics simulations and rigorously taking into account ionic correlations, we show how introducing a secondary site with a specific chemical structure in the backbone of PEO can greatly enhance the lithium conductivity of such concentrated electrolytes. In addition, we demonstrate how results based on the Nernst-Einstein equation can be highly misleading in the concentrated regime. We identify PEO-based carbonate and sulfonyl variants that respectively allow for significant ion dissociation, and high cation transference number.