  A primordial magnetic field with the strength in the 1-10 pG range can
resolve the tension between different measurements of the Hubble constant and
provide an explanation for the excess opacity in the 21 cm line at redshift
$15<z<20$, if it is present during the recombination and reionization epochs.
This field can also survive in the voids of the large-scale Structure in the
present day universe. We study the sensitivity reach of the gamma-ray technique
for measurement of such relatively strong cosmological magnetic field using
deep exposure(s) of the nearest hard spectrum blazar(s) with CTA telescopes. We
show that the gamma-ray measurement method can sense the primordial magnetic
field with a strength of up to $10^{-11}$~G. Combination of the cosmic
microwave background and gamma-ray constraints can thus sense the full range of
possible cosmological magnetic fields to confirm or rule out their relevance to
the problem of the origin of cosmic magnetic fields, as well as their influence
on recombination and reionization epochs.
