A concept of bypass rewiring is introduced, and random bypass rewiring is analytically and numerically investigated with simulations. Our results show that bypass rewiring makes networks robust against removal of nodes including random failures and attacks. In particular, random bypass rewiring connects all nodes except the removed nodes on an even degree infinite network and makes the percolation threshold 0 for arbitrary occupation probabilities. In our example, the even degree network is more robust than the original network with random bypass rewiring, while the original network is more robust than the even degree networks without random bypass. We propose a greedy bypass rewiring algorithm which guarantees the maximum size of the largest component at each step, assuming which node will be removed next is unknown. The simulation result shows that the greedy bypass rewiring algorithm improves the robustness of the autonomous system of the Internet under attacks more than random bypass rewiring.