The ray-mapping method has been widely used for designing freeform illumination lenses. However, in non-paraxial or off-axis situations, it remains challenging to obtain an integrable ray-mapping, often requiring a complex iterative correction process for the initial mapping. To address this challenge, we propose an extended ray-mapping method that incorporates differentiable ray-tracing into the design pipeline of the ray-mapping method. This enables accurate surface construction according to ray-mapping and efficient shape correction based on irradiance distribution. The proposed method involves two optimization stages. In the first stage, the freeform surface is preliminarily optimized to closely match the optimal transport mapping. The obtained freeform surface is then further optimized in the second stage to minimize the divergence between the target and simulated irradiance distributions. Additionally, the mean curvature of the freeform surface is also constrained in the second stage to facilitate the fabrication of the final freeform surface. Non-paraxial illumination lenses and off-axis illumination lenses have been designed using the proposed method within ten minutes, and simulations demonstrate that the approach is effective and robust.