Putting on Tefillin after Davening:[1]

In a case that one is wearing Tefillin after Davening, such as if he did not have Tefillin when he Davened, then he is to wear the Tefillin by Mincha, or by any other time of the day.

What is one to say while wearing the Tefillin?[2] One is to say any Mizmor[3] while wearing the Tefillin [and is not required to specifically recite Shema]. [Nonetheless, the custom is to recite Shema when wearing Tefillin, as stated in Halacha A in Q&A?]

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[1] Admur 66/11; Siddur Admur [Letter 22 Raskin]; M”A 66/12; Ketzos Hashulchan /3; Siddur Admur regarding Rabbeinu Tam says to say Shema with the Tefillin, however earlier [in letter 17 Raskin] Admur simply writes to learn with the Tefillin and does not mention the need to say a Mizmor.

[2] Admur ibid; Siddur ibid; M”A ibid; Siddur Admur regarding Rabbeinu Tam says to say Shema with the Tefillin, however earlier [in letter 17 Raskin] Admur simply writes to learn with the Tefillin and does not mention the need to say a Mizmor. Likewise in 34/5 he does not mention a requirement to say Shema with Rabbeinu Tam and that simply if one wants to do so he may.

Other opinions: Some Poskim rule that one is to recite specifically the first two Parshiyos of Shema when he wears the Tefillin, even if he has already recited it [without Tefillin]. [Levush 58/2] See Levushei Mordechai 3/5 in name of Maharam Shick 23 that from the Poskim who write one is to recite Shema with Rabbeinu Tam [see footnotes below] it is implied that just like there is Eiduis Sheker to recite Shema without Tefillin, so too it is Eidus Sheker to wear Tefillin without saying Shema.

[3] Vetzaruch Iyun as to why a Mizmor is required to be said

[4] Siddur Admur regarding Rabbeinu Tam

[5] M”A 25/6 in name of Sifri that these two Parshiyos are an obligation; Levush 58/2; This certainly applies according to those Poskim that rule that all four Parshiyos are an obligation. Now, although if it is already past the time of Shema there is room to learn that they are no longer obligated to say Shema with the Tefillin, as rules Admur 66/11 and Siddur Admur [letter 22 Raskin] and M”A 66/12 that one who is wearing Tefillin after Davening is to say with them a Mizmor. Nevertheless, they are to say it as perhaps this only applies to one who already said Shema without Tefillin due to lack of choice and not to one who is saying Shema for the first time. Furthermore, regarding Rabbeinu Tam Admur in Siddur [and see also Admur 34/5] rules that the custom is to say Shema. Thus initially one should always say Shema when wearing Tefillin. See Levushei Mordechai 3/5 in name of Maharam Shick 23 that from the Poskim who write one is to recite Shema with Rabbeinu Tam [see footnotes above] it is implied that just like there is Eiduis Sheker to recite Shema without Tefillin, so too it is Eidus Sheker to wear Tefillin without saying Shema.

[6] M”B 70/20