It is a common feature of vector-like extensions of the electroweak sector to have near degenerate states, such as electroweak doublets. In simplified models, it is usually assumed that these have decay widths saturated by two-body channels. As a consequence, experimental searches can be done focusing on only one of the states of the doublet. Taking as an example case the light exotic electroweak doublet present in the Minimal Composite Higgs Model, we show that including three-body decays in the pair production process makes this separation unfeasible, since both states of the doublet will be present and contribute significantly to the signal. In addition, by recasting present searches in multileptonic channels, with a simplified cut-and-count analysis, a relevant increase in discovery reach or exclusion potential is obtained; this indeed motivates a more detailed analysis. This study shows how an inclusive search strategy, taking into account both the near degeneracy and the presence of three-body decays, will have greater discovery power and be more natural from a model building perspective.