With the development of intelligent manufacturing, more and more nonstandard parts are used in high-precision assembly. The robotic assembly method based on attractive region in environment (ARIE) has been proven to have good performance in the high-precision assembly under the limitation of robot system accuracy or sensing accuracy. However, for the assembly of nonstandard parts, especially nonconvex parts, the existing ARIE-based strategy lacks a targeted design. In the assembly process, the nonconvex structure may cause blocking problems, which will lead to assembly failure when using the strategy. In order to solve this problem, this paper proposes a new assembly method by using the geometric features of constraint region based on the concept of ARIE. Specifically, first, when using the ARIE-based classic strategy, the reasons for the possible blocking problem in the assembly of a class of nonconvex axisymmetric parts are analyzed in detail. Second, a multi-step sliding strategy is proposed based on the theory of ARIE to solve the possible blocking problem in the assembly process. Third, impedance control is used to enable the peg to achieve the desired compliant motion in the proposed strategy. The improvement in the success rate of the proposed method is verified by the comparison experiment of small clearance peg-in-hole assembly, where the structure of the peg is nonconvex and axisymmetric.