Given Ntot applications of a unitary operation, parametrized by an unknown phase, a phase-estimation protocol on a large-scale fault-tolerant quantum system can reduce the standard deviation of an estimate of the phase from scaling as O[1/Ntot] to scaling as O[1/Ntot]. Owing to the limited resources available to near-term quantum devices, protocols that do not entangle probes have been developed. Their mean absolute error scales as O[log(Ntot)/Ntot]. Here, we propose a two-step protocol for near-term phase estimation, with an improved error scaling. Our protocol's first step produces several low-standard-deviation estimates of θ, within θ's parameter range. The second step iteratively homes in on one of these estimates. Our protocol achieves a mean-absolute-error scaling of O[log(logNtot)/Ntot] and a root-mean-square-error scaling of O[logNtot/Ntot]. Furthermore, we demonstrate a reduction in the constant scaling factor and the required circuit depths. This allows our protocol to outperform the asymptotically optimal quantum-phase-estimation algorithm for realistic values of Ntot
