Traditional hyperspectral imaging (HI) systems are constrained by a limited depth of field (DoF), necessitating refocusing for any out-of-focus objects. This requirement not only slows down the imaging speed but also complicates the system architecture. It is challenging to trade off among speed, resolution, and DoF within an ultra-simple system. While some studies have reported advancements in extending DoF, the improvements remain insufficient. To address this challenge, we propose a novel, to our knowledge, differentiable framework that integrates an extended DoF (E-DoF) wave propagation model and an achromatic hyperspectral reconstructor powered by deep learning. Through rigorous experimental validation, we have demonstrated that the compact HI system is capable of snapshot capturing of high-fidelity images with an exceptional DoF reaching approximately 5 m, marking a significant improvement of over three orders of magnitude. Additionally, the system achieves over 90% spectral accuracy without aberration, nearly doubling the accuracy performance of existing methods. An asymmetric freeform surface design is introduced for diffractive optical elements, enabling dual functionality with design freedom and E-DoF. The sparse prior conditions for spatial texture and spectral features of hyperspectral cubic data are integrated into the reconstruction network, effectively mitigating texture blurring and chromatic aberration. It foresees that the optimal strategy for achromatic E-DoF can be adopted into other optical systems such as polarization imaging and depth measurement.