Jump errors easily occur on the discontinuity of the wrapped phase because of the misalignment between wrapped phase and fringe order in fringe projection profilometry (FPP). In this paper, a phase unwrapping method that avoids jump errors is proposed for FPP. By building two other staggered wrapped phases from the original wrapped phase and dividing each period of fringe order into three parts, the proposed generalized tripartite phase unwrapping (Tri-PU) method can be used to avoid rather than compensatorily correct jump errors. It is suitable for the phase unwrapping method assisted by fringe order with a basic wrapped phase and fringe order, no matter which method is used to recover them. The experimental results demonstrate the effectiveness and generality of the proposed method, which is simple to implement and superior to measure complex objects with sharp edges.