Small vegetated patches greatly reduce urban surface temperature during a summer heatwave in Adelaide, Australia

•Cooling benefits by urban vegetation are greater at land unit than the suburb scale.•Vegetation provides greater cooling benefits to suburbs further from the coast.•Private green spaces are disproportionally important for urban heat mitigation.•Yards and trees are an overlooked asset for localised climate change adaptation.•People can adapt to heat when and where needed the most; during heatwaves and near home.