With the development of the new generation global navigation satellite system (GNSS), Chinese compass navigation systems also utilize binary offset carrier (BOC) modulation to improve the inter-operability of satellite navigation systems. The main shortcoming of BOC modulation is ambiguous in search of a multi-peaked auto-correlation function (ACF). In this paper, an improved unambiguous acquisition algorithm based on compensated correlation reconstructed technique (CCRT) is proposed for the arbitrary-order BOC, which is a new modulation technique for the next generation satellite navigation and positioning systems. The key technique used in this paper is to generate local reference signals with different shape vectors. By recombining of the piecewise correlation functions of the step-shape coded symbol (SCS), a reconstructed correlation function (RCF) that has only one peak is obtained, and the energy loss is compensated by the ACF. In the proposed algorithm, different shape vectors are employed for different BOC modulation signals. The simulation results demonstrate that the ambiguity problem is completely eliminated in the proposed algorithm. Compared with traditional acquisition methods, the proposed algorithm has an outstanding detection performance. Moreover, the CCRT can be utilized for arbitrary order BOC modulation signals.