  Applying in-plane uniaxial pressure to strongly correlated low-dimensional
systems has been shown to tune the electronic structure dramatically. For
example, the unconventional superconductor Sr$_2$RuO$_4$ can be tuned through a
single Van Hove singularity which results in a strong enhancement of both
$T_\text{c}$ and $H_\text{c2}$. Out-of-plane ($c$ axis) uniaxial pressure is
expected to tune the quasi-two-dimensional structure even more strongly,
causing it to approach two Van Hove singularities simultaneously. Here we
achieve a record value of $3.2$ GPa compression along the $c$ axis of
Sr$_2$RuO$_4$. Although the rise in $H_\text{c2}$ shows that we are indeed
approaching the van Hove points, $T_\text{c}$ is suppressed, a result that
contradicts expectations based on simple two-dimensional models. As a first
attempt to take the third dimension into account, we present three-dimensional
calculations in the weak interaction limit, and discuss the extent to which
they are consistent with observation. Our experimental results highlight the
importance of out-of-plane effects in low-dimensional systems in general and
provide new constraints on theories of the pairing interaction in
Sr$_2$RuO$_4$.
