Many storage channels admit reading and rewriting of the content at a given cost. We consider rewritable channels with uniform write noise and a hidden state which models the unknown characteristics of the memory cell. In addition to mitigating the effect of the write noise, rewrites can help the write controller obtain a better estimate of the hidden state. We present two coding strategies, each of which yields a lower bound on the rewrite capacity. We show that the second strategy is asymptotically optimal as the number of rewrites gets large.