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 Sr2RuO4 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 Sr2RuO4. 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 Sr2RuO4. In the superconductor Sr2RuO4, in-plane strain is known to enhance both the superconducting transition temperature Tc and upper critical field Hc2, but the effect of out-of-plane strain has not been studied. Here, the authors find that Hc2 is enhanced under out-of-plane strain, but Tc unexpectedly decreases.