Head movement poses a significant challenge in brain positron emission tomography (PET) imaging, resulting in image artifacts and tracer uptake quantification inaccuracies. Effective head motion estimation and correction are crucial for precise quantitative image analysis and accurate diagnosis of neurological disorders. Hardware-based motion tracking (HMT) has limited applicability in real-world clinical practice. To overcome this limitation, we propose a deep-learning head motion correction approach with cross-attention (DL-HMC++) to predict rigid head motion from one-second 3D PET raw data. DL-HMC++ is trained in a supervised manner by leveraging existing dynamic PET scans with gold-standard motion measurements from external HMT. We evaluate DL-HMC++ on two PET scanners (HRRT and mCT) and four radiotracers (18F-FDG, 18F-FPEB, 11C-UCB-J, and 11C-LSN3172176) to demonstrate the effectiveness and generalization of the approach in large cohort PET studies. Quantitative and qualitative results demonstrate that DL-HMC++ consistently outperforms state-of-the-art data-driven motion estimation methods, producing motion-free images with clear delineation of brain structures and reduced motion artifacts that are indistinguishable from gold-standard HMT. Brain region of interest standard uptake value analysis exhibits average difference ratios between DL-HMC++ and gold-standard HMT to be 1.2 plus-minus 0.5% for HRRT and 0.5 plus-minus 0.2% for mCT. DL-HMC++ demonstrates the potential for data-driven PET head motion correction to remove the burden of HMT, making motion correction accessible to clinical populations beyond research settings. The code is available at https://github.com/maxxxxxxcai/DL-HMC-TMI.