Diffusion models deliver state-of-the-art generative performance across diverse modalities but remain computationally expensive due to their inherently iterative sampling process. Existing training-free acceleration methods typically improve numerical solvers for the reverse-time ODE, yet their effectiveness is fundamentally constrained by the inefficiency of the underlying sampling trajectories. We propose A-FloPS (Adaptive Flow Path Sampler), a principled, training-free framework that reparameterizes the sampling trajectory of any pre-trained diffusion model into a flow-matching form and augments it with an adaptive velocity decomposition. The reparameterization analytically maps diffusion scores to flow-compatible velocities, yielding integration-friendly trajectories without retraining. The adaptive mechanism further factorizes the velocity field into a linear drift term and a residual component whose temporal variation is actively suppressed, restoring the accuracy benefits of high-order integration even in extremely low-NFE regimes. Extensive experiments on conditional image generation and text-to-image synthesis show that A-FloPS consistently outperforms state-of-the-art training-free samplers in both sample quality and efficiency. Notably, with as few as $5$ function evaluations, A-FloPS achieves substantially lower FID and generates sharper, more coherent images. The adaptive mechanism also improves native flow-based generative models, underscoring its generality. These results position A-FloPS as a versatile and effective solution for high-quality, low-latency generative modeling.