As we have learned, there is a disagreement among the Rishonim about whether tefillin should be worn on Chol Hamoed. Shulchon Aruch rules that they should not be worn and Rema disagrees. All agree that the tefillin of Rabeinu Tam should not be worn on Chol Hamoed. Since the Zohar writes very strongly that tefillin should not be worn on Chol Hamoed, even those who do not adhere to his words agree that an extra pair should not be worn. In addition, wearing Rabeinu Tam tefillin is doubly questionable: 1) it is possible that tefillin should not be worn at all on Chol Hamoed, and 2) the halacha may follow Rashi’s opinion.
סימן לד ס"ק כ ביאורים ומוספים דרשו 31