The surface temperature‐mediated change in cloud properties, referred to as the cloud feedback, continues to dominate the uncertainty in climate projections. A larger number of contemporary global climate models (GCMs) project a higher degree of warming than the previous generation of GCMs. This greater projected warming has been attributed to a less negative cloud feedback in the Southern Ocean. Here, we apply a novel “double decomposition method” that employs the “cloud radiative kernel” and “cloud regime” concepts, to two data sets of satellite observations to decompose the interannual cloud feedback into contributions arising from changes within and shifts between cloud morphologies. Our results show that contributions from the latter to the cloud feedback are large for certain regimes. We then focus on interpreting how both changes within and between cloud morphologies impact the shortwave cloud optical depth feedback over the Southern Ocean in light of additional observations. Results from the former cloud morphological changes reveal the importance of the wind response to warming increases low‐ and mid‐level cloud optical thickness in the same region. Results from the latter cloud morphological changes reveal that a general shift from thick storm‐track clouds to thinner oceanic low‐level clouds contributes to a positive feedback over the Southern Ocean that is offset by shifts from thinner broken clouds to thicker mid‐ and low‐level clouds. Our novel analysis can be applied to evaluate GCMs and potentially diagnose shortcomings pertaining to their physical parameterizations of particular cloud morphologies. Plain Language Summary Climate models project that rising levels of greenhouse gas emissions will raise Earth's global mean surface air temperature. However, the amount of warming to be expected remains highly uncertain. A main underlying cause for this uncertainty is the representation of atmospheric clouds in these models. In particular, the response of clouds to global warming, known as the “cloud feedback” over the Southern Ocean has been linked to higher levels of projected warming. To address the uncertainty, we apply a novel method to two satellite data sets. This method teases apart contributions from changing cloud properties such as their horizontal spatial coverage and their ability to reflect sunlight, to the cloud feedback, and then further dissects each of these contributions into those arising from changes occurring both within groups of cloud types and changes occurring between these groups. Results suggest that a general shift from thick storm‐ track clouds to thinner oceanic low‐level clouds exacerbates warming in the Southern Ocean but are partially counterbalanced by cloud thickening in part due to stronger near‐surface winds in the same region. Our novel methods and results can potentially be used to evaluate climate models and aid in tracing their shortcomings to poorly represented physical processes.