We study a low-energy effective field theory (EFT) describing the NN systemin which all exchanged particles are integrated out. We show that fitting theresidue of the 3S1 amplitude at the deuteron pole, rather than the 3S1effective range, dramatically improves the convergence of deuteron observablesin this theory. Reproducing the residue ensures that the tail of the deuteronwave function, which is directly related to NN scattering data via analyticcontinuation, is correctly reproduced in the EFT at next-to-leading order. Therole of multi-nucleon-electroweak operators which produce deviations fromeffective-range theory can then be explicitly separated from the physics of thewave function tail. Such an operator contributes to the deuteron quadrupolemoment, mu_Q, at low order, indicating a sensitivity to short-distance physics.This is consistent with the failure of impulse approximation calculations in NNpotential models to reproduce mu_Q. The convergence of NN phase shifts in theEFT is unimpaired by the use of this new expansion.