In this paper, via the generalized beam formulation, we evaluate the scintillation index for higher-order general beams propagating through the weak atmospheric turbulence. The investigated higher-order beam types are cos-Gaussian, cosh-Gaussian and annular beams. The scintillation indices of these beams are plotted with respect to propagation length, source size and order of the beam. According to our graphical outputs, higher-order beams have less on-axis scintillation index than zero-order beams at longer propagation distances. The on-axis scintillation exhibits a slight increase when the order of even-ordered beams is made higher. Moreover, our study on the source size variation shows that, for most of the source size range, cos-Gaussian beams have a lower scintillation index than the other beams.