This paper is concerned with the long-time dynamics of an epidemic model whose diffusion and reaction terms involve nonlocal effects described by suitable convolution operators, and the epidemic region is represented by an evolving interval enclosed by the free boundaries in the model. In Wang and Du (2022J. Differ. Eqn. 327 322–81), it was shown that the model is well-posed, and its long-time dynamical behaviour is governed by a spreading-vanishing dichotomy. The spreading speed was investigated in a subsequent work of Wang and Du (2023 Discrete Contin. Dyn. Syst. 43 121–61), where a threshold condition for the diffusion kernels J1 and J2 was obtained, such that the asymptotic spreading speed is finite precisely when this condition is satisfied. In this paper, we examine the case that this threshold condition is not satisfied, which leads to accelerated spreading; for some typical classes of kernel functions, we determine the precise rate of accelerated expansion of the epidemic region by constructing delicate upper and lower solutions.