The projector augmented wave (PAW) method is a powerful numerical algorithm that serves as a backend, enabling efficient density functional theory (DFT) calculations through the smoothing of valence electronic descriptions. Although it is mainly used in conjunction with plane-wave basis for solid-state systems, its generality permits the combination with other types of basis functions. In the previous study, we proposed a scheme to incorporate the PAW method into the conventional quantum chemical DFT implementation based on Gauss-type function (GTF) basis (Xiong et al., J. Chem. Theory Comput. 2017, 13, 3236-3249). The potentially high usability of the GTF-based PAW method, referred to as GTF-PAW, was previously shown, while its implementation was limited to the local density approximation (LDA). Here, we present a development of two technical extensions in this method toward practical DFT calculations. The GTF-PAW-based formulation and implementation to raise the level of the functional treatment to the generalized gradient approximation (GGA) is presented for improving reliability. In addition, we attempt to use the uniform mesh grid for DFT's quadrature in place of the conventional Becke grid, which was previously used. With the test calculations performed on illustrative molecules, it is confirmed that the conventional approach to implement GGA within GTF basis code can be straightforwardly integrated into the GTF-PAW method, allowing for the numerically stable treatment of the gradients of density. It is demonstrated that the uniform mesh grid can be used as an efficient numerical quadrature approach, which may be advantageous for handling larger systems.