The construction of a borehole correction library for the multi-component array induction tool in deviated boreholes involves extensive 3D forward modeling and typically requires significant computational resources. To address this challenge, this paper proposes an efficient algorithm for the library construction based on the 3D finite volume method (FVM) and contraction high-order Born approximation (CHBA). First, the electromagnetic (EM) field solution region is divided into two symmetric subregions based on the symmetry of the correction library model and the EM field. Numerical solution on a single subregion, combined with the symmetry boundary extension technique, enhances the efficiency of the 3D numerical simulation. Second, three reference mud conductivities are selected based on the mud conductivity range, and the CHBA is applied to calculate the EM responses at all mud conductivity nodes rapidly. Third, the number of forward simulation operations is further reduced by exploiting the principle that models with different frequencies and different formation conductivities have equivalent EM responses. Numerical experiments demonstrate the correctness and feasibility of the proposed algorithm. Compared to conventional 3D modeling, the proposed algorithm achieves approximately a 20-fold speedup in library construction, effectively reducing computational resources and time consumption.