We present a simple randomized algorithmic framework for connected\ud facility location problems. The basic idea is as follows:\ud We run a black-box approximation algorithm for the unconnected\ud facility location problem, randomly sample the clients, and open the\ud facilities serving sampled clients in the approximate solution.\ud Via a novel analytical tool, which we term core detouring,\ud we show that this approach significantly improves over the\ud previously best known approximation ratios for several NP-hard\ud network design problems. For example, we reduce the approximation\ud ratio for the connected facility location problem from 8.55 to\ud 4.00 and for the single-sink rent-or-buy problem from 3.55 to\ud 2.92. \ud The mentioned results can be derandomized at the expense of a slightly worse approximation ratio. \ud The versatility of our framework is demonstrated by devising\ud improved approximation algorithms also for other related problems