In this paper, we have constructed a bottom-up holographic model for the color superconductivity (CSC) of the Yang-Mills theory with including the higher derivative corrections which allow us to study the CSC phase with the color number $N_c\ge 2$. First, we consider the CSC phase transition in the context of Einstein-Gauss-Bonnet (EGB) gravity. We analyze the Cooper pair condensate in the deconfinement and confinement phases which are dual to the planar GB-RN-AdS black hole and GB-AdS soliton, respectively, where the backreaction of the matter part is taken into account. By examining the breakdown of the Breitenlohner-Freedman bound in the background of the planar GB-RN-AdS black hole, we find that the positive GB coupling parameter $\alpha >0$ leads to a lower upper bound of the color number in comparison to Einstein gravity where the CSC phase for $N_c\ge 2$ is not realized. But, with the $\alpha <0$ case it is possible to observe the Cooper pair condensate for $N_c\ge 2$ with the reasonable magnitude of $\alpha$. This is confirmed and the corresponding phase diagram is found by solving numerically the equations of motion for the gravitational system. In addition, we show that the CSC phase disappears in the confinement phase for the magnitude of $\alpha$ below a certain value which means that beyond that value it might lead to the breakdown region of the EGB gravity in investigating the CSC phase. However, the CSC phase transition occurring with $N_c\ge 2$ requires the magnitude of the GB coupling parameter to be rather large. As a result, the GB term would no longer be considered as the correction and it also violates the boundary causality bound. We resolve this problem by including additionally the higher derivative correction for the Maxwell electrodynamics and the nonminimal coupled Maxwell field.