The Scholars mention that one must stop talking when the Imam is speaking, not necessarily after the adhan or when he gives Salam.
Shaikh Ibn ul-‘Uthaymeen rahimahullah said:
“The silence is obligatory in the condition the Imam is giving the khutbah. So what is before the khutbah, even after the Imam comes, it is not obligatory to be silent. And between the two khutbahs, it is not obligatory to be silent” [Sharh Saheeh Bukhari (3/555)]
Shaikh Yahya ibn Ali al-Hajoori hafidhuhallah said:
” Chapter: Speaking when the Imam sits after the Salam and between the two khutbahs and after completing the Khutbah is allowed
…The Khutbah includes recitation of the something from the Quran. As for during the silence of the Khateeb when he sits after the Salam or when he is sitting between the two khutbahs or after completing the Khutbah, then the statement that speech is disliked during those times whether by the Imam or those behind the imam has no evidence for it” [Ahkaam al-Jumu’ah pg. 269]
The general principle for speaking during the Khutba is as Imam Zuhri rahimahullah said:
فَخُرُوجُ الإِمَامِ يَقْطَعُ الصَّلاَةَ وَكَلاَمُهُ يَقْطَعُ الْكَلاَم.
The imam coming out stops prayer and his speaking stops conversation
[Reported by Malik in Muwatta, Musannaf ibn Abi Shaybah, and others. Also ascribed to Sa’eed ibn Musayyib with a Saheeh chain]
So when he speaks it is not allowed to speak.
And Allah Knows Best
Faisal Ibn Abdul Qaadir Ibn Hassan