Context. Current observational evidence reveals that fast radio bursts (FRBs) exhibit bandwidths ranging from a few dozen MHz to several GHz. Traditional FRB searches primarily employ matched filter methods on time series collapsed across the entire observational bandwidth. However, with modern ultrawideband receivers featuring gigahertz-scale observational bandwidths, this approach may overlook a significant number of events. Aims. We investigate the efficacy of sub-banded searches for FRBs, whereby we look for bursts within limited portions of the bandwidth. The aim of these searches is to enhance the significance of FRB detections by mitigating the impact of noise outside the targeted frequency range, thereby improving signal-to-noise ratios. Methods. We conducted a series of Monte Carlo simulations for the 400-MHz bandwidth Parkes 21-cm multi-beam (PMB) receiver system and the Parkes Ultra-Wideband Low (UWL) receiver, simulating bursts down to frequency widths of about 100 MHz. Additionally, we performed a complete reprocessing of the high-latitude segment of the High Time Resolution Universe South survey (HTRU-S) of the Parkes-Murriyang telescope using sub-banded search techniques. Results. Simulations reveal that a sub-banded search can enhance the burst search efficiency by 67−42+133% for the PMB system and 1433−126+143% for the UWL receiver. Furthermore, the reprocessing of HTRU led to the confident detection of 18 new bursts, nearly tripling the count of FRBs found in this survey. Conclusions. These results underscore the importance of employing sub-banded search methodologies to effectively address the often modest spectral occupancy of these signals.