In recent years, a range of detailed palaeoclimatic reconstructions have shown indications of a connection between changes in solar activity and Earth's climate. However, a process that may explain such a relationship has yet to be reliably demonstrated. One proposed mechanism concerns the theoretical link between the solar-modulated galactic cosmic ray (GCR) flux and Earth's cloud cover. Several microphysical pathways have been suggested which could account for this relationship, including (i) changes in the concentration of cloud condensation nuclei by an ion-mediated growth of aerosol particles, and (ii) the accumulation of charge at cloud boundaries predicted to influence the ability of cloud droplets to grow by scavenging processes. This work uses an original epoch-superpositional (composite) methodology to present new indications of statistically significant relationships between the rate of GCR flux (measured from neutron monitor sites across the globe) and changes in the rate of numerous large-scale atmospheric parameters (derived from satellite, reanalysis and weather station datasets) over daily timescales. Attempts are made to reproduce the observed anomalous atmospheric responses within a general circulation model. The results strongly suggest that some of the observed anomalies may be causally related to cloud changes and, furthermore, provide some support for possible links between the GCR flux, cloud anomalies and atmospheric responses. This relationship appears to be sensitive to changes in the rate of GCR flux and is also likely constrained by internal conditions.