We develop an efficient evaluation scheme for the singular Fourier coefficients in solving electromagnetic scattering by body of revolution (BOR). For the singular modal Green's function (MGF), the scheme first evaluates the integrals over generating arc segments analytically and then distinguishes the singular part from the regular part for the integral over the angle in the revolution direction. This allows us to avoid the approximate calculation for the elliptical integral and the separated singular part is exactly evaluated in a closed form. For the singular MGF's derivatives generated from the gradient of the Green's function, we subtract the kernels with more similar singular integrands in the vicinity of singularity so that the kernels are better regularized. The procedure differs from the existent Glisson's method in several aspects and numerical experiments show that the scheme is simpler in implementation and more efficient for calculation.