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This article is focused on a new procedure concerning a more accurate identification of the occurrence time of an impending major earthquake (EQ). Specifically, we first recapitulate that, as was recently shown [P. Varotsos et al., Communications in Nonlinear Science and Numerical Simulation 125 (2023) 107370], natural time analysis of seismicity supplemented with the non-additive Tsallis entropy S q leads to a shortening of the time window of an impending major EQ. This has been shown for the Tohoku mega-EQ of magnitude M9 that occurred in Japan on 11 March 2011, which is the largest event ever recorded in Japan. Here, we also show that such a shortening of the time window of an impending mainshock can be achieved for major, but smaller EQs, of the order of M8 and M7. In particular, the following EQs are treated: the Chiapas M8.2 EQ, which is Mexico's largest EQ for more than a century that took place on 7 September 2017 near the coast of Chiapas state in Mexico, the 19 September 2017 M7.1 EQ that occurred within the Mexican flat slab, and the M7.1 Ridgecrest EQ on 6 July 2019 in California. [ABSTRACT FROM AUTHOR]