[1] The Poskim [see Admur 34/5; Michaber 34/2 in 2^nd custom; Maharil 137] do not make mention of any such prohibition, or act of piety, and hence we can learn from their lack of mention that refraining from speech is not necessary; Admur in Siddur writes “Wear Tefillin of Rabbeinu Tam in your house” which implies we do not suspect for an interval in-between. [Siddur Raskin footnote 71] The Rebbe was accustomed to walk to his room and only then put on Rabbeinu Tam soon after Davening, [but not immediately afterwards]. [Reply from Rabbi L.Y. Groner] The Maggid of Mezritch told his son, the Malach Reb Avraham, “Between the Tefillin of Rashi and Rabbeinu Tam you are to eat a little and then learn”; Piskeiy Teshuva’s 34/4 footnote 27 writes one may even initially make a small Hefsek in-between.

The reason: As in any event an interval has taken place between the blessing over Rashi before Davening and the current wearing, and hence according to Halacha the blessing no longer counts. [Minchas Elazar 1/25] Now, although the Chabad custom is to proximate the wearing of Rabbeinu Tam to Rashi as much as possible, perhaps this is simply to negate a long interval, and not an interval of short speech, as writes Piskeiy Teshuvos ibid.

[2] Shulchan Shlomo 34 brought in Shevach Hayakar p. 54

The Rebbe’s opinion: As stated above, the Rebbe states that one is to wear Rabbeinu Tam immediately after davening in order to connect it with the blessing over Rashi, as one cannot compare the interval of prayer to the interval of mundane matters. [Likkutei Sichos 21/358; printed in Shulchan Menachem 1/33; Shaareiy Halcha Uminhag 1/72; Sefer Haminhagim p. 38 English edition] The Rebbe however makes no mention of avoiding even speech until one does so. [Siddur Raskin ibid]

[3] The reason: One is to have in mind by the blessing over Rashi to include the Tefillin of Rabbeinu Tam and hence one is to wear the Tefillin of Rabbeinu Tam without any interval, even of speech, in order to include it within the blessing of Rashi. [Shulchan Shlomo ibid; See however Minchas Elazar ibid that certainly according to Halacha the prayer is an interval and hence cannot include Rabbeinu Tam. Perhaps one must conclude that there is a spiritual advantage in approximating the wearing of Rabbeinu Tam to the blessing, despite the fact that the blessing is legally invalid. Vetzaruch Iyun.]