It is disputed amongst Poskim as to whether one is to put on Tefillin on Chol Hamoed. Practically, the Ashkenazi custom throughout all these provinces is to don the Tefillin with a blessing, saying the blessing in private. Nevertheless it is best to don the Tefillin without a blessing due to the rule of “Safek Brachos Lehakel”. [This is the Ashkenazi custom that is followed by many until today. However some Ashkenazim are accustomed today to not wear Tefillin during Chol Hamoed. According to Kabala it is forbidden to wear Tefillin during Chol Hamoed. The Sefaradi and Chassidic custom is not to put on Tefillin at all during Chol Hamoed. This is likewise the Chabad custom. In Eretz Yisrael the widespread custom amongst all communities, including Ashkenaz, is not to wear Tefillin during Chol Hamoed.]
What is one to do if he accidently wore Tefillin during Chol Hamoed?
A Bochur once told the Rebbe he wears Tefillin during Chol Hamoed, as rules Admur in his Shulchan Aruch. The Rebbe stated that he needs an atonement for doing so and he went on to prescribe him an atonement.
 This dispute dates back to the Rishonim. Some rule one is to wear Tefillin on Chol Hamoed being that the “sign” of resting from work is not applicable by Chol Hamoed and it is this sign which is the cause for one not donning them on Shabbos and Yom Tov. [Rosh; Rama 31/2] Others however rule that the “sign” which is the cause for not donning Tefillin on Yom Tov is the particular Mitzvos of that Holiday, such as Lulav, and Chametz, which are applicable also by Chol Hamoed and thus the Tefillin should not be donned then. [Raavd; Tosafus; Rashba; Michaber 31/2; Arizal brought in Kaf Hachaim 31/6; Gr”a in Maase Rav 174]
Is a blessing recited: Even according to those which do obligate Tefillin there are opinions which hold that it should not be done with a blessing. [Admur ibid; Taz 31]
 Admur ibid; Rama 31/2
The reason: The blessing is said in private in order to show respect also for the dissenting opinion regarding the blessing and to show that there is some “sign” also on Chol Hamoed. [ibid]
Why doing so does not involve Baal Tosef or belittiling of the “sign” even according to the stringent opinion: Even according to the opinion which forbids donning Tefillin, those which don Tefillin in order to suspect for the opinion which does obligate it, are not considered transgressing the law of “do not add” or belittling the “sign” of Yom Tov even according to the stringent opinion. The reason for this is because they are not intending to do so for certain for the sake of the Mitzvah, but rather out of doubt that perhaps it is a Mitzvah, and it is only forbidden due to Baal Tosif and a belittling of the “Oas” when done for certain for the sake of a Mitzvah. Based on this it seems that donning the Tefillin is the proper way to act as there is nothing to lose by putting on the Tefillin even according to the opinion which forbids it as explained in the footnote above. However it seems that it is a Mitzvah to publicize to those which do put them on that they may not do so for certain for the Mitzvah but rather only out of doubt.
 Aruch Hashulchan 31/4; This follows the ruling of the Gr”a ibid
 See Kaf Hachaim 31/6; Mishmeres Shalom 33/8; Beis Rebbe p. 32
 Michaber 31/2
 This was one of the initiations of the Alter Rebbe, that Chassidim not wear Tefillin during Chol Hamoed. [Beis Rebbe p. 32]
 Os Chaim Veshalom 31/1