Conversational contextual bandits elicit user preferences by occasionally querying for explicit feedback on key-terms to accelerate learning. However, there are aspects of existing approaches which limit their performance. First, information gained from key-term-level conversations and arm-level recommendations is not appropriately incorporated to speed up learning. Second, it is important to ask explorative key-terms to quickly elicit the user's potential interests in various domains to accelerate the convergence of user preference estimation, which has never been considered in existing works. To tackle these issues, we first propose ``ConLinUCB", a general framework for conversational bandits with better information incorporation, combining arm-level and key-term-level feedback to estimate user preference in one step at each time. Based on this framework, we further design two bandit algorithms with explorative key-term selection strategies, ConLinUCB-BS and ConLinUCB-MCR. We prove tighter regret upper bounds of our proposed algorithms. Particularly, ConLinUCB-BS achieves a regret bound of $O(d\sqrt{T\log T})$, better than the previous result $O(d\sqrt{T}\log T)$. Extensive experiments on synthetic and real-world data show significant advantages of our algorithms in learning accuracy (up to 54\% improvement) and computational efficiency (up to 72\% improvement), compared to the classic ConUCB algorithm, showing the potential benefit to recommender systems.