An implicit–explicit Euler scheme in temporal direction is employed to discretize a partial integro-differential equation, which arises in pricing options under jump-diffusion process. Then the semi-discretized equation is approximated in space by the Sinc–Galerkin method with exponential accuracy. Meanwhile, the domain decomposition method is incorporated to handle the non-smoothness of the payoff function, and the improved fast Gauss transform is applied to accelerate the evaluation of the jump integral term. An effective preconditioner is proposed for solving the resulting dense Toeplitz-related systems by the preconditioned generalized minimum residual (GMRES) method. Numerical tests are performed to illustrate the efficiency of the proposed algorithm.