Although over centennial and greater timescales solar variability may be one of the most influential climate forcing agents, the extent to which solar activity influences climate over shorter time periods is poorly understood. If a link exists between solar activity and climate, it is likely via a mechanism connected to one (or a combination) of the following parameters: total solar irradiance (TSI), ultraviolet (UV) spectral irradiance, or the galactic cosmic ray (GCR) flux. We present an analysis based around a superposed epoch (composite) approach focusing on the largest TSI increases and decreases (the latter occurring in both the presence and absence of appreciable GCR reductions) over daily timescales. Using these composites we test for the presence of a robust link between solar activity and cloud cover over large areas of the globe using rigorous statistical techniques. We find no evidence that widespread variations in cloud cover at any tropospheric level are significantly associated with changes in the TSI, GCR or UV flux, and further conclude that TSI or UV changes occurring during reductions in the GCR flux are not masking a solar-cloud response. However, we note the detectability of any potential links is strongly constrained by cloud variability.