5G-Advanced and likely 6G will support a new low-power wake-up signal (LP-WUS) enabling low-power devices, equipped with a complementary ultra low-power receiver to monitor wireless traffic, to completely switch off their main radio. This orthogonal frequency-division multiplexed (OFDM) signal will emulate an on-off keying (OOK) modulation to enable very low-energy envelope detection at the receiver. Higher rate LP-WUS, containing multiple OOK symbols within single OFDM symbol, will be generated using the time-domain pulse multiplexing of discrete Fourier transform spread (DFT-s-) OFDM. In this context, this paper presents a comprehensive signal design framework for DFT-s-OFDM-based OOK generation. General properties of subcarrier coefficients are derived demonstrating that only DFT of the bits needs to be computed online and repeated over the band before applying appropriate frequency-domain processing. The conventional approach of generating rectangular-like OOK waveforms is then addressed by a combination of pre-DFT bit-spreading and post-DFT processing; and the least-squares (LS) method from Mazloum and Edfors, proposed for 5G LP-WUS and also Ambient-IoT, is shown to be implementable as such. Even though aesthetically pleasing and of independent interest, rectangular-like OOK waveforms are not optimal for 5G LP-WUS scenarios due to their limited robustness to channel frequency-selectivity and timing offset, and so shaping methods for spreading the OOK spectrum and concentrating the OOK symbol energy are analyzed and shown to improve the bit error rate performance under practical conditions.