Alzheimer's disease (AD) is a progressive neurodegenerative disorder that leads to irreversible cognitive decline in memory and communication. Early detection of AD through speech analysis is crucial for delaying disease progression. However, existing methods mainly use pre-trained acoustic models for feature extraction but have limited ability to model both local and global patterns in long-duration speech. In this letter, we introduce a Dual-Stage Time-Context Network (DSTC-Net) for speech-based AD detection, integrating local acoustic features with global conversational context in long-duration recordings.We first partition each long-duration recording into fixed-length segments to reduce computational overhead and preserve local temporal details.Next, we feed these segments into an Intra-Segment Temporal Attention (ISTA) module, where a bidirectional Long Short-Term Memory (BiLSTM) network with frame-level attention extracts enhanced local features.Subsequently, a Cross-Segment Context Attention (CSCA) module applies convolution-based context modeling and adaptive attention to unify global patterns across all segments.Extensive experiments on the ADReSSo dataset show that our DSTC-Net outperforms state-of-the-art models, reaching 83.10% accuracy and 83.15% F1.