Abstract 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 point, resulting in strong enhancement of both T c and H c2 . Out-of-plane ( c axis) uniaxial pressure is expected to tune the quasi-two-dimensional structure even more strongly, by pushing it towards two Van Hove points simultaneously. Here, we achieve a record uniaxial stress of 3.2 GPa along the c axis of Sr 2 RuO 4 . H c2 increases, as expected for increasing density of states, but unexpectedly T c falls. As a first attempt to explain this result, we present three-dimensional calculations in the weak interaction limit. We find that within the weak-coupling framework there is no single order parameter that can account for the contrasting effects of in-plane versus c -axis uniaxial stress, which makes this new result a strong constraint on theories of the superconductivity of Sr 2 RuO 4 .