Dai et al. [Multiscale Model. Simul. 18 (2020)] proposed a gradient flow model and a numerical scheme for ground state calculations in Kohn-Sham density functional theory. It is a feature that orthonormality of all wave functions can be preserved automatically during the simulation which makes such a method attractive towards simulations for large scale systems. In this paper, two extensions are proposed for further improving the efficiency of the method. The first one is a linearization of the original nonlinear scheme. It is shown analytically that both the orthonormality of wave functions and the decay of the total energy can be preserved well by this linear scheme, while a significant acceleration can be observed from the numerical experiments due to the removal of an iteration process in the nonlinear scheme. The second one is the introduction of the adaptivity in the algorithm both temporally and spatially — i.e. an $h$-adaptive mesh method is employed to control the total amount of mesh grids, and an adaptive stop criterion in time propagation process is designed based on an observation that total energy always decays much faster at the beginning. Plenty of numerical experiments successfully demonstrate effectiveness of our method.