An n-dimensional folded hypercube FQ_n is an attractive variance of an n-dimensional hypercube Q_n, which is obtained by a standard hypercube with some extra edges established between its vertices. An n-dimensional folded hypercube (FQ_n for short) for any odd n is known to be bipartite. In this paper, let f be a faulty vertex in FQ_n. It has been shown that (1) Every edge of FQ_n−{f} lies on a fault-free cycle of every even length l with 4≤l≤2ⁿ−2 where n≥3; (2) Every edge of FQ_n−{f} lies on a fault-free cycle of every odd length l with n+1≤l≤2ⁿ−1, where n≥2 is even. In terms of every edge lies on a fault-free cycle of every odd length in FQ_n−{f}, our result extend the result of Cheng et al. (2013) where odd cycle length up to 2ⁿ−3.