The cracks in swelling clay soils at sufficiently high water content can be presented as the superposition of interblock- and interaggregate-capillary-crack networks. The objective of this work is the generalization of an earlier proposed model describing the hydraulic conductivity of a soil matrix with only interaggregate capillary cracks. The generalized model describes the hydraulic conductivity of a soil with capillary cracks of both types. For the case of vertical hydraulic conductivity the generalization is based on the available results relative to the geometrical and hydraulic properties of capillary crack networks. The contribution of interblock capillary cracks to the hydraulic conductivity of a soil at a given pressure head changes with depth even for a given matrix structure, contrary to the contributions of interaggregate capillary cracks and soil matrix. For numerical estimates and validation of the model we use literature data on the variation of water content in different soil layers, evaporation, subsidence, and crack volume during a drying period in the course of a laboratory lysimeter experiment. The numerical estimates indicate the essential and even prevailing contribution of capillary cracks of both types to hydraulic conductivity at sufficiently small pressure heads and their negligible contribution to the water retention of clay soil. The validation is based on a comparison between two independent estimates of the summary hydraulic conductivity of the soil matrix and interaggregate capillary cracks. One of these estimates is found as a difference between the total hydraulic conductivity of the soil and the hydraulic conductivity of the interblock-capillary-crack network. Another estimate is found by the model describing the hydraulic conductivity of a soil matrix with only interaggregate capillary cracks. The results of the comparison validate the feasibility of the model showing that it is in agreement with the available data of the lysimeter experiment.