Linear stability analysis is carried out for the boundary layer under a wide range of Mach numbers (4<Mae<16) and wall temperature ratios (0.1<Tw/Taw<1.0), and for the first time, the characteristics of the third mode are analyzed under different Mae and Tw/Taw. It is found that the unstable third mode appears at Mae=10 with an adiabatic wall, and its growth rate first increases and then decreases with increasing Mae. As the wall becomes colder, the unstable third mode is enhanced and appears at lower Mae. The transition-onset variations are investigated by the eN method. As Mae reaches 14, the third mode begins to promote the transition. The transition-onset Reynolds number of the third mode can be reduced by 11.4% compared to that of the second mode at Mae=16. The transition-onset momentum thickness Reynolds number Reθt varies significantly under different Mae and Tw/Taw. At 4<Mae<10, Reθt is around 800–2000 and increases with increasing Tw/Taw significantly. At 10<Mae<16, Reθt rises to 2000–3800 and is less sensitive to the change of Tw/Taw. Therefore, it is necessary to consider the effects of Mae and Tw/Taw on Reθt in transition prediction modeling for hypersonic boundary layers.