  We report on a comprehensive reinterpretation of the existing cross-section
data for elastic electron-proton scattering obtained by the initial-state
radiation technique, resulting in a significantly improved accuracy of the
extracted proton charge radius. By refining the external energy corrections we
have achieved an outstanding description of the radiative tail, essential for a
detailed investigation of the proton finite-size effects on the measured
cross-sections. This development, together with a novel framework for
determining the radius, based on a regression analysis of the cross-sections
employing a polynomial model for the form factor, led us to a new value for the
charge radius, which is $(0.870 \pm 0.014_\mathrm{stat.}\pm 0.024_\mathrm{sys.}
\pm 0.003_\mathrm{mod.})\,\mathrm{fm}$.
