We prove that enhanced entanglement percolation via lattice transformation is possible even if the new lattice is more poorly connected in that (i) the coordination number (a local property) decreases, or (ii) the classical percolation threshold (a global property) increases. In searching for protocols to transport entanglement across a network, it seems reasonable to try transformations that increase connectivity. In fact, all examples that we are aware of violate both conditions (i) and (ii). One might therefore conjecture that all good transformations must violate them. Here we introduce a method, partial entanglement swapping, and use it to construct a counterexample that satisfies conditions (i) and (ii). The example lowers the threshold, relative to all known protocols, of the amount of initial entanglement required for deterministic long-range entanglement. This result shows that a transformation may not be rejected on the basis of satisfying conditions (i) or (ii). Both the result and the method constitute steps toward answering basic questions, such as whether there is a minimum amount of local entanglement required to achieve long-range entanglement.