Data engineering often requires accuracy (utility) constraints on results, posing significant challenges in designing differentially private (DP) mechanisms, particularly under stringent privacy parameter $ε$. In this paper, we propose a privacy-boosting framework that is compatible with most noise-adding DP mechanisms. Our framework enhances the likelihood of outputs falling within a preferred subset of the support to meet utility requirements while enlarging the overall variance to reduce privacy leakage. We characterize the privacy loss distribution of our framework and present the privacy profile formulation for $(ε,δ)$-DP and Rényi DP (RDP) guarantees. We study special cases involving data-dependent and data-independent utility formulations. Through extensive experiments, we demonstrate that our framework achieves lower privacy loss than standard DP mechanisms under utility constraints. Notably, our approach is particularly effective in reducing privacy loss with large query sensitivity relative to the true answer, offering a more practical and flexible approach to designing differentially private mechanisms that meet specific utility constraints.