Several theoretical methods have been developed to approximate prevalence and threshold of epidemics on networks. Among them, the recurrent dynamic message-passing (rDMP) theory offers a state-of-the-art performance by preventing the echo chamber effect in network edges. However, the rDMP theory was derived in an intuitive ad-hoc way, lacking a solid theoretical foundation and resulting in a probabilistic inconsistency flaw. Furthermore, real-world networks are clustered and full of local loops like triangles, whereas rDMP is based on the assumption of a locally tree-like network structure, which makes rDMP potentially inefficient on real applications. In this work, for the recurrent-state epidemics, we first demonstrate that the echo chamber effect exits not only in edges but also in local loops, which rDMP-like method can not avoid. We then correct the deficiency of rDMP in a principled manner, leading to the natural introduction of new higher-order dynamic messages, extending rDMP to handle local loops. By linearizing the extended message-passing equations, a new epidemic threshold estimation is given by the inverse of the leading eigenvalue of a matrix named triangular non-backtracking matrix. Numerical experiments conducted on synthetic and real-world networks to evaluate our method, the efficacy of which is validated in epidemic prevalence and threshold prediction tasks. In addition, our method has the potential to speed up the solution of the immunization, influence maximization, and robustness optimization problems in the networks.