Does the 144 hour visa free transit apply if I will be going from Shanghai to Taipei?

I have not applied for an entry permit because I have no time as my flight is next week. I am wondering if the 144 hour free transit applies to my situation. I wish to travel From Spain to Shanghai to attend the fair for 6 days then I want to go to Taipei for another 5 days, then go back to Shanghai, spend one more night and fly back to Spain.