Electron acceleration by a circularly polarized Hermite–Gaussian (HG) laser beam in the plasma has been investigated theoretically for the different transverse electromagnetic (TEM) mode indices (m, n) as (0, 1), (0, 2), (0, 3), and (0, 4). HG laser beam possesses higher trapping force compared with a standard Gaussian beam owing to its propagation characteristics during laser–electron interaction. A single-particle simulation indicates a resonant enhancement in the electron acceleration with HG laser beam. We present the intensity distribution for different TEM modes. We also analyze the dependence of beam width parameter on electron acceleration distance, which effectively influences the electron dynamics. Electron acceleration up to longer distance is observed with the lower modes. However, the higher electron energy gain is observed with higher modes at shorter distance of propagation.