Epoxy/clay nanocomposites are synthesized using clay modified with the organic modifier N,N-dimethyl benzyl hydrogenated tallow quaternary ammonium salt (Cloisite 10A). The purpose is to investigate the influence of the clay concentration on the nanostructure, mainly on the free-volume properties and the interfacial interactions, of the epoxy/clay nanocomposite. Nanocomposites having 1, 3, 5 and 7.5 wt. % clay concentrations are prepared using the solvent-casting method. The dispersion of clay silicate layers and the morphologies of the fractured surfaces in the nanocomposites are studied using X-ray diffraction (XRD) and scanning electron microscopy (SEM), respectively. The observed XRD patterns reveal an exfoliated clay structure in the nanocomposite with the lowest clay concentration (≤1 wt. %). The ortho-positronium lifetime (τ3), a measure of the free-volume size, as well as the fractional free volume (fv) are seen to decrease in the nanocomposites as compared to pristine epoxy. The intensity of free positron annihilation (I2), an index of the epoxy–clay interaction, decreases with the addition of clay (1 wt. %) but increases linearly at higher clay concentrations. Positron age-momentum correlation measurements are also carried out to elucidate the positron/positronium states in pristine epoxy and in the nanocomposites. The results suggest that in the case of the nanocomposite with the studied lowest clay concentration (1 wt. %), free positrons are primarily localized in the epoxy–clay interfaces, whereas at higher clay concentrations, annihilation takes place from the intercalated clay layers.