There are multiple ways to control infectious diseasesvaccination and drugs such as antibiotics or anti-virals form part of the pharmaceutical approach, however another route is to stop people infecting each other. This can be done either through general efforts to reduce epidemiologically relevant contacts, or through a more targeted attempt to trace the contacts of known cases who can then be isolated or treated. The impact of this kind of contact tracing is a priori likely to depend strongly on the network of contacts linking people together. In this paper, we develop new mathematical and computational techniques to model the impact of clustering: the probability that any two contacts of a given individual are also linked to each other in the network, creating triangles. Often, and for intuitively understandable reasons, the presence of clustering increases the efficacy of contact tracing, however we show that in the regime of highly effective contact tracing sometimes the opposite is true.