For extremely large-scale arrays (XL-arrays), the discrete Fourier transform (DFT) codebook, conventionally used in the far-field, has recently been employed for near-field beam training. However, most existing methods rely on the line-of-sight (LoS) dominant channel assumption, which may suffer degraded communication performance when applied to the general multi-path scenario due to the more complex received signal power pattern at the user. To address this issue, we propose in this paper a new hybrid learning-and-optimization-based beam training method that first leverages deep learning (DL) to obtain coarse channel parameter estimates, and then refines them via a model-based optimization algorithm, hence achieving high-accuracy estimation with low computational complexity. Specifically, in the first stage, a tailored U-Net architecture is developed to learn the non-linear mapping from the received power pattern to coarse estimates of the angles and ranges of multi-path components. In particular, the inherent permutation ambiguity in multi-path parameter matching is effectively resolved by a permutation invariant training (PIT) strategy, while the unknown number of paths is estimated based on defined path existence logits. In the second stage, we further propose an efficient particle swarm optimization method to refine the angular and range parameters within a confined search region; in the meanwhile, a Gerchberg-Saxton algorithm is used to retrieve multi-path channel gains from the received power pattern. Last, numerical results demonstrate that the proposed hybrid design significantly outperforms various benchmarks in terms of parameter estimation accuracy and achievable rate, yet with low computational complexity.