Background: Fragment yields exhibit a strong odd-even staggering (OES). This OES has been experimentally observed in different fragmentation reactions with different projectile-target combinations. However, the experimental data are still scarce for fragments close to drip lines and the origin of this OES is not well understood. Purpose: More experimental data are needed to explore the origin of this OES in fragment yields and to validate fragmentation reaction models, especially for nuclei close to the drip lines. To study the pronounced OES near the proton drip line, we measured the yields of Tz=−1 and Tz=−3/2 nuclei over a wide range of mass number. Methods: The combination of a fragment separator and a storage ring at the Heavy Ion Research Facility in Lanzhou has been used to measure the yields of Tz=−1 and Tz=−3/2 fragments, produced by 58Ni projectiles impinging on a beryllium target at an energy of about 463 MeV/nucleon. Results: A very strong OES is observed in the measured yields of both Tz=−1 and Tz=−3/2 fragments. Our experimental data demonstrate that the shell structure has a significant impact on the magnitude of this OES. A comparison of different fragmentation reaction data indicates that this OES is almost independent of the projectile-target combinations and the fragmentation energy between 140 and 650 MeV/nucleon. Conclusions: Our study reveals that the OES of fragment yields originates mainly from the OES of particle-emission threshold energies, which is very close to the OES of fragment yields when the Coulomb barrier is considered in particle-emission threshold energies.