BKK-CAN-WUH-SFO

Hello, I am having trouble getting an answer from China Southern Airlines or the Chinese Embassy on this. I am flying China Southern from Bangkok to Guangzhou with a 6-hour layover. Then I fly Guangzhou to San Francisco with a 1 hour 20-minute "stopover" in Wuhan. I am on a USA passport. Do I need to get entry permit before the trip or am I safe with the 24-hour free transit even though I have two stops in China? Any help would be greatly appreciated.