The layers of the atmosphere that were most influenced by the transition from atmosphere III to the modern atmosphere were the troposphere and the stratosphere.
The troposphere is the layer of the atmosphere closest to the Earth's surface and it is where most of the Earth's weather occurs. It is also where the majority of the Earth's air mass resides. In the transition from atmosphere III to the modern atmosphere, the troposphere experienced significant changes in the composition of gases, specifically the increase in the amount of oxygen and the decrease in the amount of carbon dioxide.
The stratosphere is the layer of the atmosphere above the troposphere and it is where the ozone layer is located. It is also where most of the Earth's atmospheric ozone is found. In the transition from atmosphere III to the modern atmosphere, the stratosphere experienced significant changes in the composition of gases, specifically the increase in the amount of ozone.
The thermosphere and mesosphere are higher layers of the atmosphere, above the stratosphere, and are less influenced by the transition from atmosphere III to the modern atmosphere. They are more related to the sun's energy and less related to the composition of gases.