Due to the inherent broadcasting nature and openness of wireless transmission channels, wireless communication systems are vulnerable to the eavesdropping of malicious attackers and usually encounter undesirable situations of information leakage. The problem may be more serious when a passive eavesdropping device is directly connected to the transmitter of a single-input single-output (SISO) system. To deal with this urgent situation, a novel IRS-assisted physical-layer secure transmission scheme based on joint transmitter perturbation and IRS reflection (JPR) is proposed, such that the secrecy of wireless SISO systems can be comprehensively guaranteed regardless of whether the reflection-based jamming from the IRS to the eavesdropper is blocked or not. Moreover, to develop a trade-off between the achievable performance and implementation complexity, we propose both element-wise and group-wise reflected perturbation alignment (ERPA/GRPA)-based IRS reflection strategies, respectively. In order to evaluate the achievable performance, we analyze the ergodic secrecy rate (ESR) and secrecy outage probability (SOP) of the SISO secure systems with the ERPA/GRPA-based JPRs, respectively. Finally, by characterizing the simulated and numerical ESR and SOP performance results, our proposed scheme is compared with the benchmark scheme of random phase-based reflection, which strongly demonstrates the effectiveness of our proposed scheme.