Causal additive models provide a tractable yet expressive framework for causal discovery in the presence of hidden variables. When unobserved backdoor or causal paths exist between two variables, their causal relationship is often unidentifiable under existing theories. We establish sufficient conditions under which causal directions can be identified in many such cases. These conditions rely on new characterizations of regression sets to determine independence among regression residuals and conditional independencies among observed variables. Building on these results, we introduce a search algorithm that incorporates these innovations and prove its soundness and completeness. Empirical evaluations demonstrate its competitive performance against state-of-the-art methods.