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A renormalizable UV model for Axion-Like Particles (ALPs) or hidden photons, that may explain the dark matter usually involves a dark Higgs field which is a singlet under the standard model (SM) gauge group. The dark sector can couple to the SM particles via the portal coupling between the SM-like Higgs and dark Higgs fields. Through this coupling, the dark sector particles can be produced in either the early universe or the collider experiments. Interestingly, not only the SM-like Higgs boson can decay into the light dark bosons, but also a light dark Higgs boson may be produced and decay into the dark bosons in a collider. In this paper, we perform the first collider search for invisible decays by taking both the Higgs bosons into account. We use a multivariate technique to best discriminate the signal from the background. We find that a large parameter region can be {probed} at the International Linear Collider (ILC) operating at the center-of-mass energy of 250 GeV. In particular, even when the SM-like Higgs invisible decay is a few orders of magnitude below the planned sensitivity reaches of the ILC and the high luminosity LHC (HL-LHC), the scenario can be probed by the invisible decay of the dark Higgs boson produced via a similar diagram. Measuring the dark Higgs boson decay into the dark sector will be a smoking gun signal of the light dark sector. A similar search of the dark sector would be expected in, e.g., Cool Copper Collider (C$^3$), Circular Electron Positron Collider (CEPC), Compact Linear Collider (CLIC) and {Future Circular electron-positron Collider (FCC-ee).