Under current LHC and dark matter constraints, the general NMSSM can have self-interacting dark matter to explain the cosmological small structure. In this scenario, the dark matter is the light singlino-like neutralino ($\chi$) which self-interacts through exchanging the light singlet-like scalars ($h_1, a_1$). These light scalars and neutralinos inevitably interact with the 125 GeV SM-like Higgs boson ($h_{SM}$), which cause the Higgs exotic decays $h_{SM} \to h_1 h_1, a_1 a_1, \chi\chi$. We first demonstrate the parameter space required by the explanation of the cosmological small structure and then display the Higgs exotic decays. We find that in such a parameter space the Higgs exotic decays can have branching ratios of a few percent, which should be accessible in the future $e^+e^-$ colliders
