Global climate models indicate that the Southern Hemispheric (SH) jet stream shifts poleward in response to CO2 forcing, but the magnitude of this shift remains highly uncertain. Here we analyse the SH jet stream response to 4×CO2 forcing in Coupled Model Intercomparison Project phase 6 (CMIP6) simulations, and find a substantially muted jet shift during winter compared with CMIP5. We suggest this muted response results from a more poleward mean jet position, consistent with a strongly reduced bias in jet position relative to the reanalysis during 1980--2004. The improved mean jet position cannot be explained by changes in the simulated sea surface temperatures. Instead, we find indications that increased horizontal grid resolution in CMIP6 relative to CMIP5 has contributed to the higher mean jet latitude, and thus to the reduced jet shift under CO2 forcing. These results imply that CMIP6 models can provide more realistic projections of SH climate change.