Although additive manufacturing (AM) continues to gain widespread adoption, the overhang problem remains a critical issue affecting printing quality. The design of self-supporting structures via topology optimisation approaches has been extensively studied. However, current optimisation research predominantly focuses on 3-axis AM machines, overlooking the more recently developed multi-axis machines. Moreover, the performance sacrifice due to overhang constraints in 3-axis AM can be significant, especially in structures with small volume fractions. To address this, we propose a two-step approach considering overhang constraints for multi-axis AM. This approach begins with a structure optimised using traditional topology optimisation. In the first step, a new optimisation problem determines printing surfaces for the given structure. If the proportion of unprintable elements isn't satisfactory, a second re-optimisation step is carried out to further reduce the unprintable proportion. Several examples demonstrate the effectiveness of the proposed approach. Notably, the significant performance sacrifice associated with the 3-axis AM approach becomes negligible when applying our multi-axis AM-based method.