Exponential resummation of the QCD finite-density Taylor series has been recently introduced as an alternative way of resumming the finite-density lattice QCD Taylor series. Unfortunately the usual exponential resummation formula suffers from stochastic bias which must be subtracted before identifying genuine higher-order contributions. In this paper, we present a new way of subtracting the stochastic bias at the level of each individual gauge configuration, up to a certain order of either the Taylor series or the cumulant expansion, by modifying the argument of the exponential. Retaining the exponential form of the resummation allows us to also calculate the phase factor of the fermion determinant on each gauge configuration. We present our results for the excess pressure, number density, and the average phase factor and show that the new results contain less stochastic bias and are in better agreement with the QCD Taylor series compared to the previous exponential resummation.