Accurate dynamic modeling is critical for autonomous racing vehicles, especially during high-speed and agile maneuvers where precise motion prediction is essential for safety. Traditional parameter estimation methods face limitations such as reliance on initial guesses, labor-intensive fitting procedures, and complex testing setups. On the other hand, purely data-driven machine learning methods struggle to capture inherent physical constraints and typically require large datasets for optimal performance. To address these challenges, this paper introduces the Fine-Tuning Hybrid Dynamics (FTHD) method, which integrates supervised and unsupervised Physics-Informed Neural Networks (PINNs), combining physics-based modeling with data-driven techniques. FTHD fine-tunes a pre-trained Deep Dynamics Model (DDM) using a smaller training dataset, delivering superior performance compared to state-of-the-art methods such as the Deep Pacejka Model (DPM) and outperforming the original DDM. Furthermore, an Extended Kalman Filter (EKF) is embedded within FTHD (EKF-FTHD) to effectively manage noisy real-world data, ensuring accurate denoising while preserving the vehicle's essential physical characteristics. The proposed FTHD framework is validated through scaled simulations using the BayesRace Physics-based Simulator and full-scale real-world experiments from the Indy Autonomous Challenge. Results demonstrate that the hybrid approach significantly improves parameter estimation accuracy, even with reduced data, and outperforms existing models. EKF-FTHD enhances robustness by denoising real-world data while maintaining physical insights, representing a notable advancement in vehicle dynamics modeling for high-speed autonomous racing.