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Higgs boson compositeness is a phenomenologically viable scenario addressing the hierarchy problem. In minimal models, the Higgs boson is the only degree of freedom of the strong sector below the strong interaction scale. We present here the simplest extension of such a framework with an additional composite spin-zero singlet. To this end, we adopt an effective field theory approach and develop a set of rules to estimate the size of the various operator coefficients, relating them to the parameters of the strong sector and its structural features. As a result, we obtain the patterns of new interactions affecting both the new singlet and the Higgs boson's physics. We identify the characteristics of the singlet field which cause its effects on Higgs physics to dominate over the ones inherited from the composite nature of the Higgs boson. Our effective field theory construction is supported by comparisons with explicit UV models.