We put forward a simpler and improved variation of a recently proposed method to overcome the signal-to-noise problem found in Monte Carlo calculations of the entanglement entropy of interacting fermions. The present method takes advantage of the approximate log-normal distributions that characterize the signal-to-noise properties of other approaches. In addition, we show that a simple rewriting of the formalism allows circumvention of the inversion of the restricted one-body density matrix in the calculation of the nth Rényi entanglement entropy for n>2. We test our technique by implementing it in combination with the hybrid Monte Carlo algorithm and calculating the n=2,3,4,⋯,10 Rényi entropies of the one-dimensional attractive Hubbard model. We use that data to extrapolate to the von Neumann (n=1) and n→∞ cases.