Recent linear stability analyses of double-diffusive convection in a laterally heated vertical slot containing water have shown that for very weak (or no) vertical salinity gradient the initial instabilities are steady, but as the salinity gradient is increased there is a transition to oscillatory instabilities. For higher Prandtl number fluids the initial instabilities in a slot with no stratification can be oscillatory or wave-like. We show that the oscillatory instabilities in water are linked to these higher Prandtl number oscillatory instabilities. The salinity gradient has a destabilizing effect on these oscillations, making them appear for Prandtl numbers where oscillatory instabilities are not possible in the absence of salinity gradients. We derive an asymptotic description for this mode of instability.