Frequency-hopped spread-spectrum transmission employing band-efficient modulations that are phase-continuous during each hop, is presented. A range of system parameters are considered, including: signaling spectrum, reception, system complexity, and performance in the presence of noise and jamming. The particular cases where the hopped modulation is minimum-shift keying (MSK), duobinary minimum-shift keying (DMSK), or tamed-FM (TFM) are studied in detail. As well, results are presented for various modulation indexes, rectangular and raised cosine pulse shapings, and a range of hop interval lengths. Sequence estimation on a hop-by-hop basis is considered. The noncoherent likelihood sequence receiver must keep all possible paths and so the computation and complexity becomes large for systems transmitting many bits per hop. Therefore, a simplified noncoherent Viterbi-like sequence estimation algorithm with reduced complexity is introduced. System performance has been evaluated in Gaussian noise, partial-band jamming and multiple-tone jamming, using bounds and a system simulation. The compact nature of the hopped spectrum when a number of bits are transmitted per hop, offers greater spectrum utilization and the prospect of improved performace in the presence of multiple-tone jamming or interference.