Frequency-hopping sequence (FHS) sets with low-hit-zone (LHZ) have Hamming correlations maintained at a low level as long as the relative time delay between different sequences are limited in a zone around the origin, and thus can be well applied in quasi-synchronous (QS) frequency-hopping multiple-access (FHMA) systems to reduce the mutual interference between different users. Moreover, the periodic partial Hamming correlation (PPHC) properties of employed LHZ-FHS sets usually act as evaluation criterions for the performances of QS-FHMA systems in practice. In this letter, a new class of LHZ-FHS sets is constructed via interleaving techniques. Furthermore, these new LHZ-FHS sets also possess optimal PPHC properties and parameters not included in the related literature.