Two-dimensional global (BiGlobal) stability in the region of the minor axis is investigated in the case of hypersonic elliptic cones with major-minor axis ratios of $2:1$ and $3:1$ at Mach number 6.0, and the BiGlobal-$e^N$ method is proposed to predict the transition location of the boundary layer. Matrix-free BiGlobal stability analysis is used to find unstable modes, including the Y-mode and the Z-mode. The growth rates in the streamwise-frequency plane for these modes are obtained. The $N_{\max\_{all}}$ factor is proposed, which represents the maximum amplification factor that all BiGlobal unstable modes can reach. Using a comparison of the $N_{\max\_{all}}$ factor with the transition location measured in a wind tunnel experiment for the $2:1$ elliptic cone, the transition prediction criterion is determined, i.e., $N_{tr} = 8.6.$ In the transition position, the amplification factors of several modes reach a level close to 8.6, which implies that none of them has the absolute superiority sufficient to cause the transition itself. Finally, the BiGlobal-$e^N$ method is employed to predict the transition location in the region of the minor axis of the $3:1$ elliptical cone. It is found that a larger major-minor axis ratio leads to stronger instability and an earlier transition.