In this work, we derive an improved triple closure for clustered pairwise models that removes two significant problems with existing closure regimes, and use this to make a systematic investigation of the impact of clustering on the efficacy of contact tracing, keeping other network and epidemiological parameters constant as appropriate. We find that, for many parameter choices, there are intuitive explanations, borne out by modelling, for the increased impact of contact tracing as clustering increases. This is not, however, a completely general result, meaning that the full implications of clustering for the efficacy of contact tracing are subtle and should be the subject of case by case investigation.