Laser-driven high-quality ion beams hold immense potential for applications in diverse fields such as tumor therapy, fast ignition, and so on. However, current experimental ion beams are often constrained by either a large energy spread or relatively low energy. In this paper, we proposed a novel scheme for generating quasi-monoenergetic proton beams by irradiating near-critical-density plasmas, which have a density gradient with a picosecond laser pulse. This approach leverages two key aspects: first, the sustained interaction between the laser pulse and the plasma enhances the duration of magnetic vortex acceleration, thereby promoting extended ion acceleration. Second, the utilization of a multi-species target facilitates the formation of a dual-peaked electric field, which leads to the accumulation of protons in the negative gradient of the accelerating phase, resulting in a quasi-monoenergetic proton beam. The two-dimensional particle-in-cell simulation reveals that by employing a laser intensity of 1.37 × 1020 W/cm2 with a pulse duration of 0.5 ps, we can achieve a carbon ion beam with an energy of 50 MeV/u, and a quasi-monoenergetic proton beam exhibiting a cutoff energy of 160 MeV/u, a peak energy of 75 MeV/u, an energy spread of 3.1 %, and an angle divergence of ∼ 3.2°. Furthermore, the quasi-monoenergetic property is corroborated in three-dimensional simulation results, underscoring the robustness and effectiveness of our proposed scheme.