Accurate tire modeling is crucial for optimizing autonomous racing vehicles, as state-of-the-art (SotA) model-based techniques rely on precise knowledge of the vehicle's parameters. Yet, system identification in dynamic racing conditions is challenging due to varying track and tire conditions. Traditional methods require extensive operational ranges, often impractical in racing scenarios. Machine learning (ML)-based methods, while improving performance, struggle with generalization and depend on accurate initialization. This paper introduces a novel on-track system identification algorithm, incorporating a neural network (NN) for error correction, which is then employed for traditional system identification with virtually generated data. Crucially, the process is iteratively reapplied, with tire parameters updated at each cycle, leading to notable improvements in accuracy in tests on a scaled vehicle. Experiments show that it is possible to learn a tire model without prior knowledge with only 30 seconds of driving data and 3 seconds of training time. This method demonstrates greater one-step prediction accuracy than the baseline nonlinear least squares (NLS) method under noisy conditions, achieving a 3.3x lower root mean square error (RMSE), and yields tire models with comparable accuracy to traditional steady-state system identification. Furthermore, unlike steady-state methods requiring large spaces and specific experimental setups, the proposed approach identifies tire parameters directly on a race track in dynamic racing environments.