The fierce weather has caused at least eight deaths and prompted advisories Saturday afternoon in New Mexico and Texas.
As thick, gray clouds covered the Southwest, forecasters said the storm would sweep across the South and toward the Atlantic coast next week, causing problems for holiday travelers.
Joe Harris, a meteorologist with the National Weather Service, said the "Nordic outbreak" will "produce a mixed bag of wily weather that will end up impacting much of the nation."
In New Mexico, authorities and residents braced for the second hit of a one-two punch that had already blanketed parts of the state with snow and freezing rain and caused a rollover accident that killed a 4-year-old girl in the eastern part of the state.
Three other storm-related deaths were reported Saturday in a crash in the Texas Panhandle involving nearly a dozen vehicles.
In California, where the storm system hit first, prompting flooding and water rescues in recent days, three deaths have been linked to the storms since Thursday, as authorities found one body near downed power lines, one man crashed his vehicle into a tree and a woman was killed when a tree fell on a parked car.
In Arizona, firefighters recovered the body of a man who was swept away by high waters Friday in the Santa Cruz River in the southern part of the state.
The storm already has affected much of the Western U.S., causing hundreds of rollover accidents and prompting officials to cancel events and close roads.
In Nevada, snow in high elevations in the rural, eastern part of the state stranded dozens of cars. No fatalities were reported and authorities got the road open again by Saturday.
In Arizona, rain came down Saturday as more than 8,000 cyclists competed in the annual El Tour de Tucson. Also, high school football games, soccer tournaments and parades were cancelled across the state.
Forecasters said parts of both California and Arizona could expect severe weather with winter storm warnings through Saturday. Weather officials said the mountains and the Antelope Valley foothills northeast of Los Angeles were under the most risk. However, they said there was only a small chance of rainstorms like those that prompted flooding in California on Thursday.
In New Mexico, it was unclear where the heaviest bands of snow would develop, meteorologist Jennifer Palucki said.
In Texas, freezing rain and cold temperatures have already hampered travel and much of the "heavy stuff" will hit south of I-20, Harris said. Several traffic accidents were reported Saturday, including the fatal crash late Friday that left several injured hurt in Vega, about 30 miles west Amarillo, and one that injured three members of singer Willie Nelson's band when their bus struck a pillar on Interstate 30 near Sulphur Springs, about 75 miles northeast of Dallas.
Contributing to this report were Associated Press writers Michelle Rindels in Las Vegas, Sudhin Thanawala in San Francisco, Paul Davenport in Phoenix, and Diana Heidgerd in Dallas.