The advancement of data privacy regulations has spurred the development of Machine Unlearning (MU), which is designed to remove the influence of sensitive data from a trained model and results in an unlearned model (ULM). Despite rapid progress in MU techniques, their vulnerabilities remain underexplored, which poses risks due to potential leakage of unlearned information. In realistic scenarios, ULMs always need to be incrementally trained with the newly collected data samples, which can lead to the consequences of recalling sensitive information if the new dataset contains similar or even the same unlearned samples. To address this issue, we devise a Durable Unlearning Enhancement (DUE) framework to avoid restoring unwanted sensitive information from incremental training data samples. The DUE framework has three key components that identify sensitive samples and suppress their gradients to update ULMs. Extensive experiments on state-of-the-art MU methods across multiple real-world datasets show that the proposed DUE framework can effectively nullify the recall of sensitive information after MU, and even improve the performance of ULMs. Consequently, our work establishes a new fundamental research direction in safe training against MU vulnerabilities.