The propagation of an intense laser pulse through an upward density-gradient plasma in a self-modulated laser wakefield acceleration (SM-LWFA) is investigated by using particle-in-cell (PIC) simulations. In the fully relativistic and kinetic PIC simulations, the relativistic and kinetic effects including Landau damping enhance the electron dephasing. This electron dephasing is the most important factor for limiting the energy of accelerated electrons. However, the electron dephasing, which is enhanced by relativistic and kinetic effects in the homogeneous plasma, can be forestalled through the detuning process arising from the longitudinal density gradient. Simulation results show that the detuning process can effectively maintain the coherence of the laser wake wave in the spatiotemporal wakefield pattern, hence considerable energy enhancement is achievable. The spatiotemporal profiles are analyzed for the detailed study on the relativistic and kinetic effects. In this paper, the optimum slope of the density gradient for increasing electron energy is presented for various laser intensities.