  We present high energy X-ray diffraction studies on the structural phases of
an optimal high-$T_c$ superconductor La$_{2-x}$Sr$_x$CuO$_{4+y}$ 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 stabilises 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 letter appear to have no direct
bearing on the electronic phase separation previously observed in the same
samples. This provides new evidence that the electronic phase separation is
determined by the overall hole concentration rather than specific Sr/O content
and concommittant structural details.
