  NGC~6558 is a low-galactic latitude globular cluster projected in the
direction of the Galactic bulge. Due to high reddening, this region presents
challenges in deriving accurate parameters, which require meticulous
photometric analysis. We present a combined analysis of near-infrared and
optical photometry from multi-epoch high-resolution images collected with
Gemini-South/GSAOI+GeMS (in the $J$ and $K_S$ filters) and HST/ACS (in the
F606W and F814W filters). We aim to refine the fundamental parameters of
NGC~6558, utilizing high-quality Gemini-South/GSAOI and HST/ACS photometries.
Additionally, we intend to investigate its role in the formation of the
Galactic bulge. We studied the impact of two differential reddening corrections
on the age derivation. When removing as much as possible the Galactic bulge
field star contamination, the isochrone fitting combined with synthetic
colour-magnitude diagrams gives a distance of $8.41^{+0.11}_{-0.10}$ kpc, an
age of $13.0\pm 0.9$ Gyr, a reddening of E($B-V$)$\,\,=0.34\pm0.02$, and a
total-to-selective coefficient R$_V = 3.2\pm0.2$ thanks to the simultaneous
near-infrared$-$Optical synthetic colour-magnitude diagram fitting. The orbital
parameters showed that NGC~6558 is confined whitin the inner Galaxy and it is
not compatible with a bar-shape orbit, indicating that it is a bulge member.
The old age of NGC~6558, combined with similar metallicity and a blue
horizontal branch in the Galactic bulge, indicates that it is part of the
moderately metal-poor globular clusters. Assembling the old and moderately
metal-poor ([Fe/H]$\,\,\sim-1.1$) clusters in the Galactic bulge, we derived
their age-metallicity relation with star formation stars at $13.6\pm0.2$ Gyr
and effective yields of $\rho=0.007\pm0.009\,\, Z_\odot$ showing a chemical
enrichment ten times faster than the ex-situ globular clusters branch.
