USA to Taiwan to Shanghai to the Philippines

Hello! So I have a US passport. I plan to travel from California to Taiwan and stay there for a few days. From Taiwan I will be flying to Shanghai for the 144-hour twov then to Manila afterwards. I know the information above states you may depart from mainland China to another country, including Hong Kong and Taiwan, but can you enter mainland China from one of these locations? Will it be possible to travel Taiwan - Shanghai - Philippines?