The earlier research showed that circularly polarized laser pulses with peak intensities in a range of 1022-1025 W/cm2 can directly accelerate and generate GeV-TeV monoenergetic electron beams with a linear energy scaling with the laser intensity. To obtain higher energy electron beams, a scheme is proposed to use an electron beam with an initial energy E0 along the laser propagation direction. This scheme can overcome the linear energy scaling with E0=0 obtained previously and enhance the beam energy by E0 folds. This is because an electron beam with an initial energy can move with the laser pulse together and therefore obtain a longer acceleration distance. Two-dimensional particle-in-cell simulation shows that this scheme is effective only for the electron beams initially with low energy on the order of MeV. With overhigh energy, electrons will miss the optimum acceleration field because the electron acceleration distance is much longer than the Rayleigh distance and the laser intensity is significantly attenuated.