Alternative binary offset carrier (AltBOC) signals can be approximated by four synchronized direct sequence spread spectrum (DSSS) signals, each pair of which is a quadrature phase shift keyed (QPSK) signal at a different frequency. Therefore, depending on the strength of an incoming AltBOC signal, an acquisition technique can reduce the mean acquisition time (MAT) by searching the four DSSS signals asynchronously; the search for each of the four DSSS signals can start at one of the evenly separated hypotheses on the two-dimensional hypothesis space. And detection sensitivity can be improved by multiple levels when different numbers of search results for the same hypothesis are combined. In this paper, we propose a fast AltBOC acquisition technique that has an asynchronous search strategy and efficiently utilizes the output of the four search results to increase the sensitivity level when sensitivity improvement is needed. We provide a complete theoretical analysis and demonstrate with numerous Monte Carlo simulations that the MAT of the proposed technique is much smaller than conventional AltBOC acquisition techniques.