Nontrivial electronic topology in strongly correlated materials is an emerging area of great interest, as it may open a way to realize entirely new quantum phases. A promising platform to explore this regime is the tetragonal, noncentrosymmetric Kondo semimetal CeRu4Sn6. It was theoretically predicted that CeRu4Sn6 could host Weyl nodes in its electronic band structure near the Fermi level [1], making it a candidate material for the newly discovered Weyl-Kondo semimetal phase [2-4]. With the discovery that CeRu4Sn6 is quantum critical without tuning [5], a natural question arises: Can the Weyl-Kondo semimetal phase nucleate out of quantum critical fluctuations and get further stabilized via tuning? Here we address this question by a study of the specific heat and magnetotransport properties of CeRu4Sn6 single crystals under hydrostatic pressure. [1] Y. Xu et al., Phys. Rev. X 7, 011027 (2017) [2] S. Dzsaber et al., Phys. Rev. Lett. 118, 246601 (2017) [3] H.-H. Lai et al., Proc. Natl. Acad. Sci. U.S.A. 115, 93 (2018) [4] S. Dzsaber et al., Proc. Natl. Acad. Sci. U.S.A. 118, e2013386118 (2021) [5] W. T. Fuhrman et al., Sci. Adv. 7, eabf9134 (2021)