We studied the phase diagram of Sr$_3$Ru$_2$O$_7$ by means of heat capacity and magnetocaloric effect measurements at temperatures as low as 0.06 K and fields up to 12 T. We confirm the presence of a new quantum critical point at 7.5 T which is characterized by a strong non-Fermi-liquid behavior of the electronic specific heat coefficient $\Delta$C/T $\sim$ -logT over more than a decade in temperature,placing strong constraints on theories of its criticality. In particular logarithmic corrections are found when the dimension d is equal to the dynamic critical exponent z, in contrast to the conclusion proposed recently [Y. Tokiwa et al., Phys. Rev. Lett. 116, 226402 (2016)]. Moreover, we achieved a clear determination of the new second thermodynamic phase adjoining the first one at lower temperatures. Its thermodynamic features differ significantly from those of the dominant phase and characteristics expected of classical equilibrium phase transitions are not observed, indicating fundamental differences in the phase formation.