Avatar reconstruction from monocular videos plays a pivotal role in various virtual and augmented reality applications. Recent methods have utilized 3D Gaussian Splatting (GS) to model human avatars, achieving fast rendering speeds with high visual quality. However, due to the independent nature of GS primitives, existing approaches often struggle to capture high-fidelity details and lack the ability to edit the reconstructed avatars effectively. To address these limitations, we propose Local Gaussian Splatting Avatar (LoGAvatar), a novel framework designed to enhance both geometry and texture modeling of human avatars. Specifically, we introduce a hierarchical Gaussian splatting framework, where local GS primitives are predicted based on sampled points from a human template model, such as SMPL. For texture modeling, we design a convolution-based texture atlas that preserves spatial continuity and enriches fine details. By aggregating local information for both geometry and texture, our approach reconstructs high-fidelity avatars while maintaining real-time rendering efficiency. Experimental results on two public datasets demonstrate the superior performance of our method in terms of avatar fidelity and rendering quality. Moreover, based on our LoGAvatar, we can edit the shape and texture of the reconstructed avatar, which inspires more customized avatar applications.