Information disorder is a challenging phenomenon that affects society at large. This phenomenon entails the diffusion of misleading, misinforming, and hateful content online. In different contexts, one aspect of the problem may prevail, but overall, this is a broad problem that requires comprehensive solutions. While each dimension of the problem (hate speech, disinformation, misinformation, etc.) requires in-depth analysis, in this paper, we look into the possibility of argument structure to provide relevant information to link these different areas of the problem. In particular, we focus on the WSF-ARG+ dataset, which consists of white supremacy forum messages annotated in terms of argument structure (premises and conclusion). There, we leverage the checkworthiness and hatefulness annotations of the argument components to obtain insights into the hatefulness of the whole message. Our results show promising insights (up to 96% F1), indicating the possibility of extending this direction in the future to tackle hateful content identification and information disorder countering.