As black-box machine learning models become more complex and are applied in high-stakes settings, the need for providing explanations for their predictions becomes crucial. Although Local Interpretable Model-agnostic Explanations (LIME) \cite{ribeiro2016should} is a widely adopted method for understanding model behavior, it suffers from instability with respect to random seeds \cite{zafar2019dlime, shankaranarayana2019alime, bansal2020sam} and exhibits low local fidelity (i.e., how the explanation explains model's local behaviors) \cite{rahnama2019study, laugel2018defining}. Our study demonstrates that this instability is caused by small sample weights, resulting in the dominance of regularization and slow convergence. Additionally, LIME's sampling approach is non-local and biased towards the reference, leading to diminished local fidelity and instability to references. To address these challenges, we propose \textsc{Glime}, an enhanced framework that extends LIME and unifies several previous methods. Within the \textsc{Glime} framework, we derive an equivalent formulation of LIME that achieves significantly faster convergence and improved stability. By employing a local and unbiased sampling distribution, \textsc{Glime} generates explanations with higher local fidelity compared to LIME, while being independent of the reference choice. Moreover, \textsc{Glime} offers users the flexibility to choose sampling distribution based on their specific scenarios.