True Digital Orthophoto Map (TDOM), a 2D objective representation of the Earth's surface, is an essential geospatial product widely used in urban management, city planning, land surveying, and related applications. However, traditional TDOM generation typically relies on a complex offline photogrammetric pipeline, leading to substantial latency and making it unsuitable for time-critical or real-time scenarios. Moreover, the quality of TDOM may deteriorate due to inaccurate camera poses, imperfect Digital Surface Model (DSM), and incorrect occlusions detection. To address these challenges, this work introduces A-TDOM, a near real-time TDOM generation method built upon On-the-Fly 3DGS (3D Gaussian Splatting) optimization. As each incoming image arrives, its pose and sparse point cloud are computed via On-the-Fly SfM. Newly observed regions are then incrementally reconstructed as additional 3D Gaussians are inserted using a Delaunay triangulated Gaussian sampling and integration and are further optimized via adaptive training iterations and learning rate, especially in previously unseen or coarsely modeled areas. With orthogonal splatting integrated into the rendering pipeline, A-TDOM can actively produce updated TDOM outputs immediately after each 3DGS update. Code is now available at https://github.com/xywjohn/A-TDOM.