Answer
In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
In this fatwa:
1- Sending blessings upon Prophet Muhammad (peace be upon him) after the adhan (the call to prayer) is firmly established in the authentic hadiths.
2- There is no text that obligates a single manner, audible or silent, for sending blessings upon Prophet Muhammad.
Answering your question about sending blessings upon Prophet Muhammad after adhan, Dar Al-Ifta Al-Misriyyah, states:
An Act of Sunnah
Sending blessings upon Prophet Muhammad (peace be upon him) after the adhan (the call to prayer) is firmly established in the authentic hadiths.
Abdullah Ibn Amr Ibn al-As (may Allah be pleased with them both) narrated that he heard Prophet Muhammad say, “When you hear the muezzin calling for prayer, repeat what he says and then invoke blessings upon me, because whoever does so will be earn a tenfold reward. Then beseech God to grant me al-Wasilah: it is a high rank in Paradise fit for only one of God’s servants, and I hope to be that person. I will intercede for whoever asks that I attain al-Wasilah.” (Muslim, Abu-Dawud, At-Tirmdhi and An-Nasa`i)
Formula of Sending Blessings upon Prophet Muhammad After the Adhan
There is no text that obligates a single manner, audible or silent, for sending blessings upon Prophet Muhammad; the matter is open and capacious.
When God Almighty legislates a matter in a general way and it may be executed and applied through various means, it must be left general and it is impermissible to restrict it to a certain manner without supporting proof.
There are reports that support the permissibility of sending blessings upon Prophet Muhammad audibly after the adhan.
Abu Ad-Darda (may Allaj be pleased with him) narrated that after hearing the adhan, Prophet Muhammad would say, “Allahuma Rab hadhihi ad-dawah at-tama was-salah al-qaimah, salli ala Muhammad wa atih sualihu yawm al-qiyamah [O God, Lord of this most perfect call and prayer that is now being established, send your blessings upon Muhammad and grant him his wish on the Day of Resurrection]” (At-Tabarani in Al-Mu’jam Al-Kabeer)
This was heard by those around him and they would repeat the muezzin’s words. Prophet Muhammad said, “Whoever says this when he hears the muezzin, it becomes incumbent upon me to intercede for him on the Day of Judgment.”
So, the matter is open and the right thing to do is to let people behave naturally. Whoever wants to send blessings upon Prophet Muhammad is free to make it in the manner and words which best appeal to him since it is important that the manner and formula resonate in his heart. It is impermissible for anyone to deny what others do as long as the matter is open and capacious.
Almighty Allah knows best.
Source: Dar Al-Ifta Al-Misriyyah