Tensor-network Born machines (TNBMs) are quantum-inspired generative models for learning data distributions. Using tensor-network contraction and optimization techniques, the model learns an efficient representation of the target distribution, capable of capturing complex correlations with a compact parameterization. Despite their promise, the optimization of TNBMs presents several challenges. A key bottleneck of TNBMs is the logarithmic nature of the loss function commonly used for this problem. The single-tensor logarithmic optimization problem cannot be solved analytically, necessitating an iterative approach that slows down convergence and increases the risk of getting trapped in one of many non-optimal local minima. In this paper, we present an improved second-order optimization technique for TNBM training, which significantly enhances convergence rates and the quality of the optimized model. Our method employs a modified Newton's method on the manifold of normalized states, incorporating regularization of the loss landscape to mitigate local minima issues. We demonstrate the effectiveness of our approach by training a one-dimensional matrix product state (MPS) on both discrete and continuous datasets, showcasing its advantages in terms of stability and efficiency, and demonstrating its potential as a robust and scalable approach for optimizing quantum-inspired generative models.