Syndromic surveillance is an effective tool for enabling the timely detection of infectious disease outbreaks and facilitating the implementation of effective mitigation strategies by public health authorities. While various information sources are currently utilized to collect syndromic signal data for analysis, the aggregated measurement of cough, an important symptom for many illnesses, is not widely employed as a syndromic signal. With recent advancements in ubiquitous sensing technologies, it becomes feasible to continuously measure population-level cough incidence in a contactless, unobtrusive, and automated manner. In this work, we demonstrate the utility of monitoring aggregated cough count as a syndromic indicator to estimate COVID-19 cases. In our study, we deployed a sensor-based platform (Syndromic Logger) in the emergency room of a large hospital. The platform captured syndromic signals from audio, thermal imaging, and radar, while the ground truth data were collected from the hospitals electronic health record. Our analysis revealed a significant correlation between the aggregated cough count and positive COVID-19 cases in the hospital (Pearson correlation of 0.40, p-value < 0.001). Notably, this correlation was higher than that observed with the number of individuals presenting with fever (ρ = 0.22, p = 0.04), a widely used syndromic signal and screening tool for such diseases. Furthermore, we demonstrate how the data obtained from our Syndromic Logger platform could be leveraged to estimate various COVID-19-related statistics using multiple modeling approaches. Aggregated cough counts and other data, such as people density collected from our platform, can be utilized to predict COVID-19 patient visits related metrics in a hospital waiting room, and SHAP and Gini feature importance-based metrics showed cough count as the important feature for these prediction models. Furthermore, we have shown that predictions based on cough counting outperform models based on fever detection (e.g., temperatures over 39°C), which require more intrusive engagement with the population. Our findings highlight that incorporating cough-counting based signals into syndromic surveillance systems can significantly enhance overall resilience against future public health challenges, such as emerging disease outbreaks or pandemics.