This manuscript introduces the convergence Epidemic Volatility Index (cEVI), a modification of the recently introduced Epidemic Volatility Index (EVI), as an early warning tool for emerging epidemic waves. cEVI has a similar architectural structure as EVI, but with an optimization process inspired by a Geweke diagnostic-type test. Our approach triggers an early warning based on a comparison of the most recently available window of data samples and a window based on the previous time frame. Application of cEVI to data from the COVID-19 pandemic data revealed steady performance in predicting early, intermediate epidemic waves and retaining a warning during an epidemic wave. Furthermore, we present two basic combinations of EVI and cEVI: (1) their disjunction cEVI + that respectively identifies waves earlier than the original index, (2) their conjunction cEVI- that results in higher accuracy. Combination of multiple warning systems could potentially create a surveillance umbrella that would result in early implementation of optimal outbreak interventions.