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Automating the research for the best neural network model is a task that has gained more and more relevance in the last few years. In this context, Neural Architecture Search (NAS) represents the most effective technique whose results rival the state of the art hand-crafted architectures. However, this approach requires a lot of computational capabilities as well as research time, which makes prohibitive its usage in many real-world scenarios. With its sequential model-based optimization strategy, Progressive Neural Architecture Search (PNAS) represents a possible step forward to face this resources issue. Despite the quality of the found network architectures, this technique is still limited in research time. A significant step in this direction has been done by Pareto-Optimal Progressive Neural Architecture Search (POPNAS), which expands PNAS with a time predictor to enable a trade-off between search time and accuracy, considering a multi-objective optimization problem. This paper proposes a new version of the Pareto-Optimal Progressive Neural Architecture Search, called POPNASv2. Our approach enhances its first version and improves its performance. We expanded the search space by adding new operators and improved the quality of both predictors to build more accurate Pareto fronts. Moreover, we introduced cell equivalence checks and enriched the search strategy with an adaptive greedy exploration step. Our efforts allow POPNASv2 to achieve PNAS-like performance with an average 4x factor search time speed-up.