Consider percolation on the Hamming cube {0,1}^n at the critical probability p_c (at which the expected cluster size is 2^{n/3}). It is known that if p=p_c(1+O(2^{-n/3}), then the largest component is of size roughly 2^{2n/3} with high probability and that this quantity is non-concentrated. We show that for any sequence eps(n) such that eps(n)>>2^{-n/3} and eps(n)=o(1) percolation at...