The storm killed at least 6 people in Taiwan and has thrust millions into the dark. Some 1.4 million homes in China's Fujian province were evacuated ahead of its landfall.
Meteorologists say this year's hurricane season will be quieter than usual. But if a Category 3 or higher hit the U.S., it would be the first time since 2005, one of the worst years on record.
The first tropical storm of the Atlantic season arrives early, as Texas, Oklahoma and other states hope to avoid a repeat of devastating tornadoes that hit earlier this week.
A cool Atlantic Ocean and a mild-to-moderate El Niño in the Pacific equatorial region have forecasters predicting only four or five Atlantic hurricanes this year.
Hurricane Odile is now a tropical storm, but it has heavy rainfall and power outages to the Baja Peninsula and surrounding areas. It could also bring new flooding to Tucson and nearby areas.
The Category 3 cyclone, the strongest on record to hit the region, slammed the tourist destination of Cabo San Lucas and will weaken as it moves up the Baja California peninsula, forecasters say.