Actually as far as i understand, NOT putting on tfillin on chol hamoed is a chumrah. People who don’t have a set minhag, are told not to start. Rabeinu Tam tfilin are never worn on chol hamoed.
When I moved back from EY to England, Rav Moshe Sternbuch told me to be matir neder (as I had the minhag from home before living in EY), and even then to put on tfilin after davening, betzin’a (i.e at home), without a brocho.