No nontrivial optimal sets of frequency-hopping sequences (FHSs) of period 2(2ⁿ - 1) for a positive integer n ≥ 2 have been found so far, when their frequency set sizes are less than their periods. In this paper, systematic doubling methods to construct new FHS sets are presented under the constraint that the set of frequencies has size less than or equal to 2ⁿ. First, optimal FHS sets with respect to the Peng-Fan bound are constructed when frequency set size is either 2ⁿ - 1 or 2ⁿ. And then, near-optimal FHS sets with frequency set size 2ⁿ - 1 are designed by applying the Chinese Remainder Theorem to Sidel'nikov sequences, whose FHSs are optimal with respect to the Lempel-Greenberger bound. Finally, a general construction is given for near-optimal FHS sets whose frequency set size is less than 2ⁿ - 1. Our constructions give new parameters not covered in the literature, which are summarized in Table 1.