  Bilayer strontium ruthenate Sr$_3$Ru$_2$O$_7$ displays pronounced non-Fermi
liquid behavior at magnetic fields around 8 T, applied perpendicular to the
ruthenate planes, which previously has been associated with an itinerant
metamagnetic quantum critical end point (QCEP). We focus on the magnetic
Gr\"uneisen parameter $\Gamma_{\rm H}$, which is the most direct probe to
characterize field-induced quantum criticality. We confirm quantum critical
scaling due to a putative two-dimensional QCEP near 7.845(5) T, which is masked
by two ordered phases A and B, identified previously by neutron scattering. In
addition we find evidence for a QCEP at 7.53(2) T and determine the quantum
critical regimes of both instabilities and the effect of their superposition.
