Speakers often have multiple ways to express the same meaning. The Uniform Information Density (UID) hypothesis suggests that speakers exploit this variability to maintain a consistent rate of information transmission during language production. Building on prior work linking UID to syntactic reduction, we revisit the finding that the optional complementizer $\textit{that}$ in English complement clauses is more likely to be omitted when the clause has low information density (i.e., more predictable). We advance this line of research by analyzing a large-scale, contemporary conversational corpus and using machine learning and neural language models to refine estimates of information density. Our results replicated the established relationship between information density and $\textit{that}$-mentioning. However, we found that previous measures of information density based on matrix verbs' subcategorization probability capture substantial idiosyncratic lexical variation. By contrast, estimates derived from contextual word embeddings account for additional variance in patterns of complementizer usage.