  We present an improved action for renormalizable effective field theories
(EFTs) of systems near the two-body unitarity limit. The ordering of EFT
interactions is constrained, but not entirely fixed, by the renormalization
group. The remaining freedom can be used to improve the theory's convergence,
to simplify its applications, and to connect it to phenomenological models. We
exemplify the method on a contact theory applied to systems of up to five
$^4$He atoms. We solve the EFT at leading order including a subleading
interaction that accounts for part of the two-body effective range. We show
that the effects of such fake range can be compensated in perturbation theory
at next-to-leading order, as long as the fake range is smaller or comparable to
the experimental effective range. These results open the possibility of using
similar improved actions for other many-body systems.
