To put it simply, mammoth habitat then was not necessarily colder than that same habitat type is now. Had mammoths survived the last glaciation, they would still have woolly coats and live in the cold temperate parts of the Northern Hemisphere just as all the extant animals they once shared it with still do.
Siberia?, the central highlands of Asia, eg Tibet, Mongolia?, Northern Sweden, Norway? And the American species, tundra areas of Alaska and Canada?