Global warming will soon reach the Paris Agreement targets of 1.5C°/2°C temperature increase. Under a business-as-usual scenario, the time to reach these targets varies widely among climate models. Using Coupled Model Intercomparison Project phase 5 and 6, we show that a 2°C near-future global warming rate is determined by Southern Ocean (SO) state closely tied with a low-level cloud (LLC) amount feedback strength during the reference (1861-1900) period; climate models with cold SO tend to accompany more low-level cloudiness and Antarctic sea ice concentration due to a strong LLC amount feedback. Consequently, initially cold SO models tend to simulate a fast near-future warming rate by absorbing more downward shortwave radiation compared to initially warm SO models because more LLC disappears due to a strong LLC amount feedback during the 2°C rise. Our results demonstrate that climate models that correctly simulate initial SO state can improve near-future projections with reduced uncertainties.