An accurate range model with acceleration, the coupling phase terms, and spatial-variant (SV) Doppler parameters are the main issues to be solved in high-speed-high-squint synthetic aperture radar (HSHS-SAR) with a curved trajectory. For these issues, an extended Omega-K (EOK) algorithm is developed in this article. The proposed EOK algorithm mainly includes the following four aspects. First, a modified range model (AMRM) considering 3-D acceleration for a curved trajectory is established. Then, the coupling between the range and azimuth direction is removed by the modified Stolt mapping (MSM). Subsequently, an improved high-order SV phase correction approach is derived to eliminate the azimuth dependence of Doppler parameters. Finally, in order to avoid zeros-padding operation, the proposed method focuses on the sub-aperture data in the range time and azimuth frequency domain through data aligning processing. The experimental results of both simulation and real data verify the effectiveness of the proposed method.