Current regression losses for tiny object detection exhibit fundamental limitations in error sensitivity control. While L1 loss maintains constant gradients regardless of error magnitude (impeding high-precision convergence), L2 loss suffers from outlier sensitivity due to its quadratic growth. Though Smooth L1 introduces threshold-based L1/L2 switching, its fixed transition boundary fails to address the critical need for adaptive gradient modulation, particularly vital for small objects where minor deviations significantly impact IoU calculations. To address the limitations of existing regression losses, we propose a Gradient-Adaptive Hyperbolic Loss (GAHL) that intrinsically couples gradient behaviors with error scales. Specifically, GAHL loss first overcomes the weak supervision of L1 near convergence through hyperbolic gradient amplification, which progressively intensifies updates for diminishing errors, achieving several times stronger gradients than L1 for sub-pixel deviations critical to IoU sensitivity. Then by replacing the rigid threshold in Smooth L1 with error-adaptive blending of L1/L2 regimes, it eliminates oscillatory optimization, reducing convergence instability on sub-16-pixel targets. Simultaneously, controlled saturation mechanics suppress large geometrically irrelevant errors, capping their gradient magnitudes at 70% of L2 loss values to mitigate outlier sensitivity. These mechanisms are unified through a learnable hyperbolic tangent operator that smoothly transitions between gradient regimes without hard thresholds. Our method achieves consistent accuracy improvements across four datasets, with  6.6% AP gains on AI-TOD, demonstrating its effectiveness in error-scale-aware optimization for tiny targets.