To achieve the extreme high data rate and extreme coverage extension requirements of 6G wireless communication, new spectrum in sub-THz (100-300GHz) and non-terrestrial network (NTN) are two of the macro trends of 6G candidate technologies, respectively. However, non-linearity of power amplifiers (PA) is a critical challenge for both sub-THz and NTN. Therefore, high power efficiency (PE) or low peak to average power ratio (PAPR) waveform design becomes one of the most significant 6G research topics. Meanwhile, high spectral efficiency (SE) and low out-of-band emission (OOBE) are still important key performance indicators (KPIs) for 6G waveform design. Single-carrier waveform discrete Fourier transform spreading orthogonal frequency division multiplexing (DFT-s-OFDM) has achieved many research interests due to its high PE, and it has been supported in 5G New Radio (NR) when uplink coverage is limited. So DFT-s-OFDM can be regarded as a candidate waveform for 6G. Many enhancement schemes based on DFT-s-OFDM have been proposed, including null cyclic prefix (NCP)/unique word (UW), frequency-domain spectral shaping (FDSS), and time-domain compression and expansion (TD-CE), etc. However, there is no unified framework to be compatible with all the enhancement schemes. This paper firstly provides a general description of the 6G candidate waveforms based on DFT-s-OFDM enhancement. Secondly, the more flexible TD-CE supporting methods for unified non-orthogonal waveform (uNOW) are proposed and discussed. Thirdly, a unified waveform framework based on DFT-s-OFDM structure is proposed. By designing the pre-processing and post-processing modules before and after DFT in the unified waveform framework, the three technical methods (NCP/UW, FDSS, and TD-CE) can be integrated to improve three KPIs of DFT-s-OFDM simultaneously with high flexibility. Then the implementation complexity of the 6G candidate waveforms are analyzed and compared. Performance of different DFT-s-OFDM enhancement schemes is investigated by link level simulation, which reveals that uNOW can achieve the best PAPR performance among all the 6G candidate waveforms. When considering PA back-off, uNOW can achieve 124% throughput gain compared to traditional DFT-s-OFDM.