Many global climate models underestimate the cloud cover and overestimate the cloud albedo, especially for low-level clouds. We determine how a correct representation of the vertical structure of clouds can fix part of this bias. We use the 1D McICA framework and focus on low-level clouds. Using LES results as reference, we propose a method based on exponential-random overlap (ERO) that represents the cloud overlap between layers and the subgrid cloud properties over several vertical scales, with a single value of the overlap parameter. Starting from a coarse vertical grid, representative of atmospheric models, this algorithm is used to generate the vertical profile of the cloud fraction with a finer vertical resolution, or to generate it on the coarse grid but with subgrid heterogeneity and cloud overlap that ensures a correct cloud cover. Doing so we find decorrelation lengths are dependent on the vertical resolution, except if the vertical subgrid heterogeneity and interlayer overlap are taken into account coherently. We confirm that the frequently used maximum-random overlap leads to a significant error by underestimating the low-level cloud cover with a relative error of about 50%, that can lead to an error of SW cloud albedo as big as 70%. Not taking into account the subgrid vertical heterogeneity of clouds can cause an additional relative error of 20% in brightness, assuming the cloud cover is correct.