To cater to users' desire for an immersive browsing experience, numerous e-commerce platforms provide various recommendation scenarios, with a focus on Trigger-Induced Recommendation (TIR) tasks. However, the majority of current TIR methods heavily rely on the trigger item to understand user intent, lacking a higher-level exploration and exploitation of user intent (e.g., popular items and complementary items), which may result in an overly convergent understanding of users' short-term intent and can be detrimental to users' long-term purchasing experiences. Moreover, users' short-term intent shows uncertainty and is affected by various factors such as browsing context and historical behaviors, which poses challenges to user intent modeling. To address these challenges, we propose a novel model called Deep Uncertainty Intent Network (DUIN), comprising three essential modules: i) Explicit Intent Exploit Module extracting explicit user intent using the contrastive learning paradigm; ii) Latent Intent Explore Module exploring latent user intent by leveraging the multi-view relationships between items; iii) Intent Uncertainty Measurement Module offering a distributional estimation and capturing the uncertainty associated with user intent. Experiments on three real-world datasets demonstrate the superior performance of DUIN compared to existing baselines. Notably, DUIN has been deployed across all TIR scenarios in our e-commerce platform, with online A/B testing results conclusively validating its superiority.