Despite achieving state-of-the-art accuracy on temporal ordering of events, neural models showcase significant gaps in performance. Our work seeks to fill one of these gaps by leveraging an under-explored dimension of textual semantics: rich semantic information provided by explicit textual time cues. We develop STAGE, a system that consists of a novel temporal framework and a parser that can automatically extract time cues and convert them into representations suitable for integration with neural models. We demonstrate the utility of extracted cues by integrating them with an event ordering model using a joint BiLSTM and ILP constraint architecture. We outline the functionality of the 3-part STAGE processing approach, and show two methods of integrating its representations with the BiLSTM-ILP model: (i) incorporating semantic cues as additional features, and (ii) generating new constraints from semantic cues to be enforced in the ILP. We demonstrate promising results on two event ordering datasets, and highlight important issues in semantic cue representation and integration for future research.