<p>We explore the phenomenological consequences of non-minimal hidden sector interactions on observable correlations in the Higgs sector, mediated through the ℤ₂-symmetric Higgs portal. Particular attention is given to non-standard momentum dependencies of the hidden sector scalar, which arise naturally in an effective field theory (EFT) framework, e.g. in Composite Scalar Dark Matter theories. We discuss the implications of such hidden sector interactions for the thermal history of the universe. We show that aspects of such non-standard momentum dependencies can be probed at future lepton colliders such as a FCC-ee, potentially also through radiative corrections. This gives rise to precision probes for regions where direct detection constraints and relic abundance can be accounted for as predicted in, e.g., Composite Scalar Dark Matter theories.</p>