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Abstract This paper is concerned with the nonlinear Klein–Gordon–Maxwell system {−Δz+V(x)z−(2ω+ϕ)ϕz=g(x,z)x∈R3,Δϕ=(ω+ϕ)z2x∈R3, $$\begin{aligned} \textstyle\begin{cases} -\Delta z+V(x)z-(2\omega +\phi )\phi z= g(x,z) & x\in \mathbb{R}^{3}, \\ \Delta \phi =(\omega +\phi )z^{2} &x\in \mathbb{R}^{3}, \end{cases}\displaystyle \end{aligned}$$ where the potential V and the primitive of g are both allowed to be sign-changing. Under more general superlinear assumptions on the nonlinearity, we obtain a new existence result of infinitely many high energy solutions by using variational methods. Some recent results in the literature are generalized and significantly improved.