Considering a Markov chain defined on a cycle, near-quadratic improvement of mixing is shown when only a subtle perturbation is introduced to the structure and non-reversible transition probabilities are used. More precisely, a mixing time of $O(n^{\frac{k+2}{k+1}})$ can be achieved by adding $k$ random edges to the cycle, keeping $k$ fixed while $n\to\infty$. The construction builds upon a biased random walk along the cycle.