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The compound Ca₃Co₂O₆ undergoes a transition into a spin-density wave (SDW) state near 24 K. Below ~10 K, this unstable SDW state coexists with a nearly- degenerate commensurate antiferromagnetic state as well as short-range magnetic order. Clear signatures of this strong magnetic disorder have been observed in the response of entropy to changing magnetic field and temperature. We performed a calorimetry study of Ca₃Co₂O₆ and Ca₃Co_1.9Zn_0.1O₆ in order to compare their entropic responses at low temperature. Our results for Ca₃Co₂O₆ reveal that ΔS(T, H) ≡ S(T, H) - S(T, H = 0) increases as either temperature or magnetic field increase. In contrast, ΔS data for Ca₃Co_1.9Zn_0.1O₆ were relatively unresponsive to changes in temperature or field, suggesting that Zn substitution may reduce the low-temperature magnetic disorder observed in Ca₃Co₂O₆. These results will be discussed within the context of two cases (Ca₃Co₂O₆ under applied pressure and Ca_2.75R_0.25Co₂O₆ (R = Dy, Lu)) in which a single magnetic ground state is stabilized.