Estimating the frequencies of multiple, closely-spaced noisy sinusoids when the data record is short is commonly accomplished by subspace-based methods such as ESPRIT, MUSIC, etc. These methods do not assume that the data are zero outside the observation interval. If we assume otherwise, the threshold SNR lowered significantly, but the price paid is unacceptable bias. Among all known unbiased estimators, the maximum-likelihood estimator (MLE) has the lowest threshold, but is computationally the most expensive. We propose a new algorithm that carries out, when needed, (i) zero-padding, and (ii) removal and re-estimation. These added steps result in a threshold SNR that is lower than that of the MLE for the examples considered herein, viz., noisy signals containing sinusoids with random parameters and up to five components. The maximum improvement in threshold was 10 dB for the two-sinusoid case. The bias of the estimates is also either equal to or lower than MLE's. Unlike the MLE, the proposed method is very much computationally feasible. © 2021 Elsevier B.V.