The Matthias rule, which is an empirical correlation between the superconducting transition temperature (Tc) and the average number of valence electrons per atom (n) in alloys and intermetallic compounds, has been used in the past as a guiding principle to search for new superconductors with higher Tc. The intermetallic compound SrBi3 (AuCu3 structure) exhibits a Tc of 5.6 K. An ab-initio electronic band structure calculation for SrBi3 predicted that Tc increases on decreasing the Fermi energy, i.e., on decreasing n, because of a steep increase in the density of states. In this study, we demonstrated that high-pressure (~ 3 GPa) and low-temperature ( < 350 °C) synthesis conditions enables the substitution of Na for about 40 at.% of Sr. With a consequent decrease in n, the Tc of (Sr,Na)Bi3 increases to 9.0 K. A new high-Tc peak is observed in the oscillatory dependence of Tc on n in compounds with the AuCu3 structure. We have shown that the oscillatory dependence of Tc is in good agreement with the band structure calculation. Our experiments reaffirm the importance of controlling the number of electrons in intermetallic compounds.