A critical task in multi-pass process planning for the five-axis machining of complicated parts is to determine the intermediate surfaces for rough machining. Traditionally, the intermediate surfaces are simply parallel Z-level planes, and the machining is of the simplest three-axis type. However, for complicated parts, this so-called Z-level method lacks flexibility and causes isolated islands on layers, which require extraneous air movements by the tool. Moreover, the in-process workpiece machined according to the Z-level method suffers from the staircase effect, which often induces unstable dynamic problems on the tool-spindle system. In this paper, we propose a new method of planning a five-axis machining process for a complicated freeform solid part. In our method, the intermediate surfaces are no longer planar but curved, and they are intrinsically influenced by the convex hull of the part. The powerful algebraic tool of geodesic distance field is utilized to generate the desired intermediate surfaces, for which collision-free five-axis machining tool paths are then planned. In addition, we propose a novel idea of alternating between the roughing and finishing machining operations, which helps improve the stiffness of the in-process workpiece. Ample physical cutting experiments are performed, and the experimental results convincingly confirm the advantages of our method.