Random walks on graphs can be slow. To speed them up, imagine that at each step instead of choosing the neighbor at random, there is a small probability $\varepsilon>0$ that we can choose it. We show that in this case, at least for graphs of bounded degree, there is a way to steer the walk so that it visits every vertex in $n^{1+o(1)}$ steps with high probability. The key to this result is a way to decompose arbitrary graphs into small-diameter pieces.