  In this paper, I will update the current status of the carbon-carbon fusion
research taking into account that after the latest analysis [Beck {\it et al.}
Eur. Phys. J. A {\bf 56}, 97 (2020), Letter to the Editor] new important
experimental and theoretical results had been published and will discuss how to
advance new THM measurements to extract the low-energy astrophysical
$S$-factors.
