copy paste from torah musings . com
By: Rabbi Ari Enkin
On Rosh Chodesh the tefillin are removed before Mussaf.[1] [2]
The second reason tefillin are removed before Mussaf is in order to add a Yom Tov flavor to the day. As Mussaf represents the Yom Tov nature of Rosh Chodesh the tefillin are removed at that time.[3] In fact, some authorities recommend removing the tefillin before the Torah reading as it also represents the Yom Tov nature of the day.[4] There are also those who recommend removing the tefillin before Hallel for the same reason.[5] Some suggest removing the tefillin after the Torah is returned to the aron kodesh.[6] Nevertheless, the halacha is not in accordance with any of these views.[7]
[8] Once the tefillin are removed they are usually just placed on a table and not properly put away and in most cases they are not even wrapped up. The reason for this is in order to minimize the interruption between the kaddish and the start of Mussaf.[9] Nevertheless, the tefillin should at least be covered before one begins Mussaf.[10]
Although this approach is normative and based on kabbalisitc considerations,[12] [13] [14] Indeed, this approach is consistent with other occasions when there is some form of an interruption before Mussaf. For example, in most congregations the rabbi delivers the Shabbat morning drasha *before* the kaddish is recited. In this way, Mussaf can commence immediately after the kaddish with no interruptions.[15]
Furthermore, leaving the tefillin exposed and unwrapped for the duration of Mussaf is unbecoming the reverence owed to them. In fact, according to halacha, whenever one removes tefillin one is to wrap and put away the shel-rosh before even removing the shel yad! There does not seem to be any good reason why this halacha should be ignored on Rosh Chodesh.[16] Folding the tefillin during the repetition of Mussaf is also not an option as one is required to listen attentively to the repetition and not engage in any other activities.[17] As such, the tefillin are left unwrapped and exposed for an exceptionally long time.[18]
Therefore, it appears to me that the ideal procedure on Rosh Chodesh would be to remove, wrap, and put away the tefillin before the kaddish is recited. Once everyone finishes putting away their tefillin the kaddish can be recited and the entire congregation can begin Mussaf at once without any interruption. As mentoned, this is conssitent with other occasions when there is an interruption before Mussaf.
[19] [20]
[1] OC 25:13; Mishna Berura 25:59; Kitzur Shulchan Aruch 10:19; Rivevot Ephraim 7:317.
[2] Mishna Berura 25:61; Shulchan Aruch Harav, OC 25:41.
[3] OC 423:4; Levush 25:13; Mishna Berura 423:10.
[4] Levush 423:4; Rema Mipano 108.
[5] Rema Mipano 108.
[6] Magen Avraham 28:30.
[7] Magen Avraham 423:6.
[8] Magen Avraham 25:30.
[9] Rambam, Hilchot Tefilla 9:13; Taamei Haminhagim, Inyanei Rosh Chodesh.
[10]
[12] Magen Avraham 423:6, Pri Megadim, EA 423:6; Mishna Berura 25:59; Kaf Hachaim, OC 25:94,95.
[13] Mishna Berura 25:59.
[14] Siddur Baal Hatanya.
[15] Mishna Berura 25:59.
[16] Mishna Berura 28:8.
[17] Mishna Berura 124:17; Igrot Moshe, OC 4:19; Tzitz Eliezer 11:10.
[18] Pri Megadim, MZ 34:2.
[19] Mishna Berura 66:35.
[20] Igrot Moshe, OC 3:4. For more on the discussion of when to remove the tefillin on Rosh Chodesh see: Rivevot Ephraim 1:283.