Contactless hand pose estimation requires sensors that provide precise spatial information and low computational complexity for real-time processing. Unlike vision-based systems, radar offers lighting independence and direct motion assessments. Yet, there is limited research balancing real-time constraints, suitable frame rates for motion evaluations, and the need for precise 3D data. To address this, we extend the ultra-efficient two-tone hand imaging method from our prior work to a three-tone approach. Maintaining high frame rates and real-time constraints, this approach significantly enhances reconstruction accuracy and precision. We assess these measures by evaluating reconstruction results for different hand poses obtained by an imaging radar. Accuracy is assessed against ground truth from a spatially calibrated photogrammetry setup, while precision is measured using 3D-printed hand poses. The results emphasize the method's great potential for future radar-based hand sensing.