Secondary instability of a streaky boundary layer under spanwise-localized free-stream vortical disturbances (FSVD) is investigated using BiGlobal instability analysis. The instability analysis is performed at different streamwise locations and at different instants in a whole time period and the results in space-time $(x-t)$ plane are taken into consideration, such that unstable modes are found in four unstable zones. According to the comparison of mode growth accumulation, we find that the contribution of secondary instability to bypass transition is much more important than T-S waves and that the center strong streak plays the key role in secondary instability. The sinuous type outer mode at center low-speed streak is the dominant unstable mode and this agrees with experimental observation. The inner mode at center high-speed streak also has high growth rate, but its growth cannot keep accumulating, resulting in its unimportant role in secondary instability. All unstable modes found in this paper appear at Fjortoft inflection points of the basic flow in gradient direction, implying that the modes have the same physical nature, i.e., they are caused by inflectional instability of the shear flow.