Conversational Query Reformulation (CQR) has significantly advanced in addressing the challenges of conversational search, particularly those stemming from the latent user intent and the need for historical context. Recent works aimed to boost the performance of CRQ through alignment. However, they are designed for one specific retrieval system, which potentially results in poor generalization. To overcome this limitation, we present a novel framework AdaCQR. By aligning reformulation models with both term-based and semantic-based retrieval systems, AdaCQR enhances the generalizability of information-seeking queries across diverse retrieval environments through a dual-phase training strategy. We also developed two effective approaches for acquiring superior labels and diverse input candidates, boosting the efficiency and robustness of the framework. Experimental evaluations on the TopiOCQA and QReCC datasets demonstrate that AdaCQR significantly outperforms existing methods, offering both quantitative and qualitative improvements in conversational query reformulation.