I add: The report appears in Sunan Abū Dāwūd, yet he did not cite it in full. In one place he merely wrote, “and he mentioned the ḥadīth,” and at the end, “and he summarized the ḥadīth,” indicating that he abridged it; consequently, Bilāl’s words, “So I put…,” and the like are absent from his version.
It seems clear that the phrase formed part of what he omitted. His chain there is as follows: Abū Tūbah al-Rabīʿ ibn Nāfiʿ narrated to us; Muʿāwiyah—meaning Ibn Sallām—narrated from Zayd, who said he heard Abū Salām state: ʿAbd Allāh al-Hawzanī related it to him.
This isnād is sound: all of its transmitters are those relied upon by Muslim, with the exception of ʿAbd Allāh al-Hawzanī—whose personal name is Ibn Laḥy—yet he himself is trustworthy and lived through both the pre-Islamic and Islamic periods.
Third corroboration: on the authority of Saʿd al-Qaraẓ, who said that the Messenger of Allah ﷺ instructed Bilāl to place his index fingers in his ears, adding, “That will make your voice carry farther.”
Ibn Mājah and al-Ṭabarānī in al-Ṣaghīr recorded it through the route of ʿAbd al-Raḥmān ibn Saʿd ibn ʿAmmār ibn Saʿd: “my father narrated to me from his father, from his grandfather.”
Al-Ḥākim also transmitted it, but with Saʿd omitted from the chain.
This isnād is weak, suffering from both weak transmitters and unknown ones; the same chain has already occurred for another report in the second category of narrations on the adhān.
Al-Ṭabarānī likewise cited it in al-Kabīr with the wording: “When you call the adhān, put your index fingers in your ears, for that will make your voice carry farther.”