Data imputation is pivotal in the analysis of incomplete time series. This study addresses the challenge of imputing consecutively missing data in time series by effectively capturing long-term temporal dependencies from incomplete segments. Two basic steps are employed to realize this idea: (1) feature representation of incomplete segments and (2) dependency learning based on the represented features. We propose a model named ORBA, which uses Overlapping Reconstruction and Biased Attention for consecutive imputation. Firstly, the biased attention mechanism is harnessed to represent the features of incomplete segments in time series. It incorporates two additional learnable vectors to enhance representation stability under diverse missing data conditions. Secondly, the overlapping reconstruction method is designed to learn temporal dependencies within and between segments. The method utilizes the represented features of one segment to reconstruct both the segment itself and its adjacent segments using distinct reconstructors, facilitating the learning of imputation while ensuring the effectiveness of representation. To evaluate the performance of ORBA, we compare it with four imputation models and three long-term forecasting models on four datasets. The experiment results demonstrate that ORBA outperforms the best baselines by about 15.1% in terms of imputation accuracy across various consecutive missing data conditions.