We study the consensus problem in a synchronous distributed system of $n$ nodes under an adaptive adversary that has a slightly outdated view of the system and can block all incoming and outgoing communication of a constant fraction of the nodes in each round. Motivated by a result of Ben-Or and Bar-Joseph (1998), showing that any consensus algorithm that is resilient against a linear number of crash faults requires $\tilde Ω(\sqrt{n})$ rounds in an $n$-node network against an adaptive adversary, we consider a late adaptive adversary, who has full knowledge of the network state at the beginning of the previous round and unlimited computational power, but is oblivious to the current state of the nodes. Our main contributions are randomized distributed algorithms that achieve almost-everywhere consensus w.h.p. against a late adaptive adversary who can block up to $εn$ nodes in each round, for a small constant $ε>0$. Our first protocol achieves binary and plurality consensus, and the second one achieves multi-value consensus. Both of our algorithms succeed in $O(\log n)$ rounds with high probability, thus showing an exponential gap to the aforementioned lower bound for strongly adaptive crash-failure adversaries, which can be strengthened to $Ω(n)$ when allowing the adversary to block nodes instead of permanently crashing them. In our algorithms each node contacts only an (amortized) constant number of peers in each communication round. We show that our algorithms are optimal up to constant (resp. sub-logarithmic) factors by proving that every almost-everywhere consensus protocol takes $Ω(\log_d n)$ rounds in the worst case, where $d$ is an upper bound on the number of communication requests initiated per node in each round. We complement our theoretical results with an experimental evaluation of the first protocol revealing a short convergence time.