We present high-energy x-ray diffraction studies on the structural phases of an optimal high-T_c superconductor La_2-xSr_xCuO_4+y tailored by co-hole-doping. This is specifically done by varying the content of two very different chemical species, Sr and O, respectively, in order to study the influence of each. A superstructure known as staging is observed in all samples, with the staging number n increasing for higher Sr dopings x. We find that the staging phases emerge abruptly with temperature, and can be described as a second-order phase transition with transition temperatures slightly depending on the Sr doping. The Sr appears to correlate the interstitial oxygen in a way that stabilizes the reproducibility of the staging phase both in terms of staging period and volume fraction in a specific sample. The structural details as investigated in this paper appear to have no direct bearing on the electronic phase separation previously observed in the same samples. This provides evidence that the electronic phase separation is determined by the overall hole concentration rather than specific Sr/O content and concomitant structural details.