Recently, 3D head avatar modeling based on 3D Gaussians has demonstrated significant advantages in rendering quality and efficiency, provided there is sufficient data. Some efforts have begun to train prior models on large datasets to develop generalizable 3D Gaussian head avatar modeling methods. Unfortunately, due to the limited expressive power of identity-shared 3D representations, the prior-based modeling often result in degenerated rendering quality. To overcome this limitation, we propose to formulate the 3D Gaussian head avatar modeling as a joint reconstruction and registration problem. Given static input images (e.g., a short mobile phone capture), we optimize two sets of 3D Gaussians: the prior-based one possesses complete animation rigging information inferred from the prior model and produces plausible modeling results, while the prior-free one is used to more freely capture the fine-grained geometric and texture details in the input images. Additionally, we simultaneously solve the registration problem between the two 3D Gaussian sets. On one hand, the registration results will provide binding information for the prior-free reconstruction to make it animatable. On the other hand, during optimization, the prior-based Gaussian can regularize the prior-free reconstruction to resist overfitting and perform good in novel expressions. Finally, we merge the parts of the prior-based reconstruction that are occluded in the input images with the prior-free reconstruction set, and then apply appropriate post-processing strategies (such as teeth enhancement) to produce a complete head avatar. We evaluated our method on the public Nersemble dataset and our own in-the-wild data. The experiments demonstrate that, under the same experimental settings, our method significantly improves modeling quality and provides better support for detailed modeling at higher resolutions.