  Over twenty years ago, Type Ia Supernovae (SNIa) [arXiv:astro-ph/9805201,
arXiv:astro-ph/9812133] observations revealed an accelerating Universe
expansion, suggesting a significant dark energy presence, often modelled as a
cosmological constant, $\Lambda$. Despite its pivotal role in cosmology, the
standard $\Lambda$CDM model remains largely underexplored in the redshift range
between distant SNIa and the Cosmic Microwave Background (CMB). This study
harnesses the James Webb Space Telescope's advanced capabilities to extend the
Hubble flow mapping across an unprecedented redshift range, from $z \approx 0$
to $z \approx 7.5$. Utilising a dataset of 231 HII galaxies and extragalactic
HII regions, we employ the $\text{L}-\sigma$ relation, correlating the
luminosity of Balmer lines with their velocity dispersion, to define a
competitive technique for measuring cosmic distances. This approach maps the
Universe's expansion over more than 12 billion years, covering 95\% of its age.
Our analysis, using Bayesian inference, constrains the parameter space $\lbrace
h, \Omega_m, w_0\rbrace = \lbrace 0.731\pm0.039, 0.302^{+0.12}_{-0.069},
-1.01^{+0.52}_{-0.29}\rbrace $ (statistical) for a flat Universe. These results
provide new insights into cosmic evolution and suggest uniformity in the
photo-kinematical properties of young massive ionizing clusters in giant HII
regions and HII galaxies across most of the Universe's history.
