The fast orthogonal search (FOS) algorithm has been shown to accurately model various types of time series by implicitly creating a specialized orthogonal basis set to fit the desired time series. When the data contain periodic components, FOS can find frequencies with a resolution greater than the discrete Fourier transform (DFT) algorithm. Frequencies with less than one period in the record length, called subharmonic frequencies, and frequencies between the bins of a DFT, can be resolved. This paper considers the resolution of subharmonic frequencies using the FOS algorithm. A new criterion for determining the number of non-noise terms in the model is introduced. This new criterion does not assume the first model term fitted is a dc component as did the previous stopping criterion. An iterative FOS algorithm called FOS first-term reselection (FOS-FTR), is introduced. FOS-FTR reduces the mean-square error of the sinusoidal model and selects the subharmonic frequencies more accurately than does the unmodified FOS algorithm.