Oil spills represent a severe threat, making early-stage thickness estimation crucial for guiding remediation efforts. Unmanned Aerial Vehicles (UAVs) are an attractive platform for environmental monitoring. However, due to their limited computation and power budgets, real-time onboard processing requires optimized algorithms or lightweight machine learning models. While the standard U-Net architecture is often too large for constrained UAV hardware, the compressed Tiny U-Net variant fits on FPGA platforms and achieves competitive estimation performance (0.79 in the metric Intersection over Union, or IoU). Despite this success, Tiny U-Net processes every radar image through the complete inference pipeline, resulting in unnecessary computation for simple cases. To address this inefficiency, we integrate an early exit feature into the Tiny U-Net architecture. We introduce an early exit branch that returns an early prediction when a compact confidence score exceeds a tunable threshold, bypassing deeper layers for high-confidence evaluations. Our experiments demonstrate that this design achieves comparable IoU to the full baseline model. Crucially, the technique is shown to reduce the average number of multiplications by up to 42% for an aggressive threshold, reducing the dynamic power consumption. Choosing a threshold that ensures extreme confidence reduces the complexity-reduction gains for an improved IoU. This early exit approach substantially improves computational efficiency in Tiny U-Net, enabling more practical deployment in UAV-based environmental monitoring systems.