The generation of smoother and shorter spiral complete coverage paths in multi-connected domains is a crucial research topic in path planning for robotic cavity machining and other related fields. Traditional methods for spiral path planning in multi-connected domains typically incorporate a subregion division procedure that leads to excessive subregion bridging, requiring longer, more sharply turning, and unevenly spaced spirals to achieve complete coverage. To address this issue, this paper proposes a novel spiral complete coverage path planning method using conformal slit mapping. It takes advantage of the fact that conformal slit mapping can transform multi-connected domains into regular disks or annuluses without the need for subregion division. Firstly, a slit mapping calculation technique is proposed for segmented cubic spline boundaries with corners. Secondly, a spiral path spacing control method is developed based on the maximum inscribed circle radius between adjacent conformal slit mapping iso-parameters. Thirdly, the spiral coverage path is derived by offsetting iso-parameters. Numerical experiments indicate that our method shares a comparable order of magnitude in computation time with the traditional PDE-based spiral complete coverage path method, but it excels in optimizing total path length, smoothness, and spacing consistency. Finally, we performed experiments on cavity milling and dry runs to compare the new method with the traditional PDE-based method in terms of machining duration and steering impact, respectively. The comparison reveals that, with both algorithms achieving complete coverage, the new algorithm reduces machining time and steering impact by 12.34% and 22.78%, respectively, compared with the traditional PDE-based method.