In this paper we propose a method for sinusoidal frequency estimation that improves upon our previously proposed LSF-based algorithm that used at most 5p candidate points, where p is the number of sinusoids present. In this paper we propose the following improvements: (i) reduced the number of candidate frequencies to at most 2p points, (ii) reduced the method's threshold to equal that of ML, and (iii) reduced the computational burden by switching to methods like ESPRIT when the SNR is above threshold. Since neither the SNR nor the threshold is known, we estimate them from the data. The proposed reduction-in-threshold step can be applied to EPUMA (proposed Qian et al.), with which we compare our results. For the well-known two-sinusoid example the proposed method has the same threshold as that of ML; ML performance is also achieved when tested on a new, three-sinusoid example. © 2020 IEEE.