We address policy learning with logged data in contextual bandits. Current offline-policy learning algorithms are mostly based on inverse propensity score (IPS) weighting requiring the logging policy to have \emph{full support} i.e. a non-zero probability for any context/action of the evaluation policy. However, many real-world systems do not guarantee such logging policies, especially when the action space is large and many actions have poor or missing rewards. With such \emph{support deficiency}, the offline learning fails to find optimal policies. We propose a novel approach that uses a hybrid of offline learning with online exploration. The online exploration is used to explore unsupported actions in the logged data whilst offline learning is used to exploit supported actions from the logged data avoiding unnecessary explorations. Our approach determines an optimal policy with theoretical guarantees using the minimal number of online explorations. We demonstrate our algorithms' effectiveness empirically on a diverse collection of datasets.