There are different ways to make the Adhaan according to the Sunnah besides the usual method.

1. To make Adhaan by pronouncing the Takbeer twice at the beginning instead of four times:

عَنْ أَبِي مَحْذُورَةَ، أَنَّ نَبِيَّ اللَّهِ صلى الله عليه وسلم عَلَّمَهُ هَذَا الأَذَانَ ‏”‏ اللَّهُ أَكْبَرُ اللَّهُ أَكْبَرُ أَشْهَدُ أَنْ لاَ إِلَهَ إِلاَّ اللَّهُ أَشْهَدُ أَنْ لاَ إِلَهَ إِلاَّ اللَّهُ أَشْهَدُ أَنَّ مُحَمَّدًا رَسُولُ اللَّهِ أَشْهَدُ أَنَّ مُحَمَّدًا رَسُولُ اللَّهِ – ثُمَّ يَعُودُ فَيَقُولُ – أَشْهَدُ أَنْ لاَ إِلَهَ إِلاَّ اللَّهُ أَشْهَدُ أَنْ لاَ إِلَهَ إِلاَّ اللَّهُ أَشْهَدُ أَنَّ مُحَمَّدًا رَسُولُ اللَّهِ أَشْهَدُ أَنَّ مُحَمَّدًا رَسُولُ اللَّهِ حَىَّ عَلَى الصَّلاَةِ – مَرَّتَيْنِ – حَىَّ عَلَى الْفَلاَحِ – مَرَّتَيْنِ ‏”‏ ‏.‏ زَادَ إِسْحَاقُ ‏”‏ اللَّهُ أَكْبَرُ اللَّهُ أَكْبَرُ لاَ إِلَهَ إِلاَّ اللَّهُ ‏”‏

Abu Mahdhura said that the Messenger of Allah (ﷺ) taught him Adhan like this:
Allah is the Greatest, Allah is the Greatest; I testify that there is no god but Allah, I testify that there is no god but Allah; I testify that Muhammad Is the Messenger of Allah, I testify that Muhammad is the Messenger of Allah, and it should be again repeated: I testify that there is no god but Allah, I testify that there is no god but Allah; I testify that Muhammad Is the Messenger of Allah, I testify that Muhammad is the Messenger of Allah. Come to the prayer (twice). Come to success (twice). Ishaq added: Allah is the Greatest, Allah is the Greatest; there Is no god but Allah.

[Saheeh Muslim no. 379]

2. Making Tarjee’ [repeating the shahatayan quietly to oneself then reciting them loudly again]

Proof:

أَنَّ عَبْدَ اللَّهِ بْنَ مُحَيْرِيزٍ، أَخْبَرَهُ – وَكَانَ، يَتِيمًا فِي حَجْرِ أَبِي مَحْذُورَةَ حَتَّى جَهَّزَهُ إِلَى الشَّامِ – قَالَ قُلْتُ لأَبِي مَحْذُورَةَ إِنِّي خَارِجٌ إِلَى الشَّامِ وَأَخْشَى أَنْ أُسْأَلَ عَنْ تَأْذِينِكَ فَأَخْبَرَنِي أَنَّ أَبَا مَحْذُورَةَ قَالَ لَهُ خَرَجْتُ فِي نَفَرٍ فَكُنَّا بِبَعْضِ طَرِيقِ حُنَيْنٍ مَقْفَلَ رَسُولِ اللَّهِ صلى الله عليه وسلم مِنْ حُنَيْنٍ فَلَقِيَنَا رَسُولُ اللَّهِ صلى الله عليه وسلم فِي بَعْضِ الطَّرِيقِ فَأَذَّنَ مُؤَذِّنُ رَسُولِ اللَّهِ صلى الله عليه وسلم بِالصَّلاَةِ عِنْدَ رَسُولِ اللَّهِ صلى الله عليه وسلم فَسَمِعْنَا صَوْتَ الْمُؤَذِّنِ وَنَحْنُ عَنْهُ مُتَنَكِّبُونَ فَظَلِلْنَا نَحْكِيهِ وَنَهْزَأُ بِهِ فَسَمِعَ رَسُولُ اللَّهِ صلى الله عليه وسلم الصَّوْتَ فَأَرْسَلَ إِلَيْنَا حَتَّى وَقَفْنَا بَيْنَ يَدَيْهِ فَقَالَ رَسُولُ اللَّهِ صلى الله عليه وسلم ‏”‏ أَيُّكُمُ الَّذِي سَمِعْتُ صَوْتَهُ قَدِ ارْتَفَعَ ‏”‏ ‏.‏ فَأَشَارَ الْقَوْمُ إِلَىَّ وَصَدَقُوا فَأَرْسَلَهُمْ كُلَّهُمْ وَحَبَسَنِي فَقَالَ ‏”‏ قُمْ فَأَذِّنْ بِالصَّلاَةِ ‏”‏ ‏.‏ فَقُمْتُ فَأَلْقَى عَلَىَّ رَسُولُ اللَّهِ صلى الله عليه وسلم التَّأْذِينَ هُوَ بِنَفْسِهِ قَالَ ‏”‏ قُلِ اللَّهُ أَكْبَرُ اللَّهُ أَكْبَرُ اللَّهُ أَكْبَرُ اللَّهُ أَكْبَرُ أَشْهَدُ أَنْ لاَ إِلَهَ إِلاَّ اللَّهُ أَشْهَدُ أَنْ لاَ إِلَهَ إِلاَّ اللَّهُ أَشْهَدُ أَنَّ مُحَمَّدًا رَسُولُ اللَّهِ أَشْهَدُ أَنَّ مُحَمَّدًا رَسُولُ اللَّهِ ‏”‏ ‏.‏ ثُمَّ قَالَ ‏”‏ ارْجِعْ فَامْدُدْ صَوْتَكَ ‏”‏ ‏.‏ ثُمَّ قَالَ ‏”‏ قُلْ أَشْهَدُ أَنْ لاَ إِلَهَ إِلاَّ اللَّهُ أَشْهَدُ أَنْ لاَ إِلَهَ إِلاَّ اللَّهُ أَشْهَدُ أَنَّ مُحَمَّدًا رَسُولُ اللَّهِ أَشْهَدُ أَنَّ مُحَمَّدًا رَسُولُ اللَّهِ حَىَّ عَلَى الصَّلاَةِ حَىَّ عَلَى الصَّلاَةِ حَىَّ عَلَى الْفَلاَحِ حَىَّ عَلَى الْفَلاَحِ اللَّهُ أَكْبَرُ اللَّهُ أَكْبَرُ لاَ إِلَهَ إِلاَّ اللَّهُ ‏”‏ ‏.‏ ثُمَّ دَعَانِي حِينَ قَضَيْتُ التَّأْذِينَ فَأَعْطَانِي صُرَّةً فِيهَا شَىْءٌ مِنْ فِضَّةٍ فَقُلْتُ يَا رَسُولَ اللَّهِ مُرْنِي بِالتَّأْذِينِ بِمَكَّةَ ‏.‏ فَقَالَ ‏”‏ قَدْ أَمَرْتُكَ بِهِ ‏”‏ ‏.‏ فَقَدِمْتُ عَلَى عَتَّابِ بْنِ أَسِيدٍ عَامِلِ رَسُولِ اللَّهِ صلى الله عليه وسلم بِمَكَّةَ فَأَذَّنْتُ مَعَهُ بِالصَّلاَةِ عَنْ أَمْرِ رَسُولِ اللَّهِ صلى الله عليه وسلم ‏.‏

‘Abdul-‘Aziz bin ‘Abdul-Malik bin Abu Mahdhurah narrated that ‘Abdullah bin Muhairiz – who was an orphan under the care of Abu Mahdhurah until he prepared him to go to Ash-Sham – informed him:
he said: “I said to Abu Mahdhurah: ‘I am going to Ash-Sham and I am afraid that I will be asked about how you say the Adhan.”‘He told me that Abu Mahdhurah said to him, I went out with a group of people and we were somewhere on the road to Hunain when the Messenger of Allah (ﷺ) was coming back from Hunain. The Messenger of Allah met us somewhere on the road and the Muadhdhin of the Messenger of Allah called the Adhan for prayer in the presence of the Messenger of Allah. We heard the voice of the Muadh’dhin and we were careless about it (the Adhan), so we started yelling, imitating and mocking it. The Messenger of Allah (ﷺ) heard us, so he sent some people who brought us to stand infront of him. He said, ‘Who is the one whose voice I heard so loud?’ The people all pointed to me, and they were telling the truth. He sent them all away, but kept me there and said to me: ‘Stand up and call the Adhan for the Prayer.’ I stood up and the Messenger of Allah taught me the Adhan himself. He Said, ‘Say: ‘Allahu Akbar, Allahu akbar, Allahu Akbar, Allahu Akbar; Ashhadu an la ilaha illallah, Ashhadu an la ilaha illallah; Ashhadu anna Muhammadan Rasulallah, Ashhadu anna Muhammadan Rasulallah (Allah is the Greatest,Allah is the Greatest, Allah is the Greatest, Allah is the Greatest; I bear witness that there is none worthy of worship except Allah, I bear witness that there is none worthy of worship except Allah; I bear witness that Muhammad is the Messenger Allah,I bear witness that Muhammad is the Messenger Allah).’ Then he said: ‘Then repeat and say in a loud voice:Ashhadu an la ilaha illallah, Ashhadu an la ilaha illallah; Ashhadu anna Muhammadan Rasulallah, Ashhadu anna Muhammadan Rasulallah; Hayya ‘alas-salah, Hayya ‘ala-salah; Hayya ‘alal-falah Hayya ‘alal-falah; Allahu Akbar, Allahu Akbar; La ilaha ill-Allah (I bear witness that there is none worthy of worship except Allah, I bear witness that there is none worthy of worship except Allah; I bear witness that Muhammad is the Messenger of Allah, I bear witness that Muhammad is the Messenger of Allah; Come to prayer, come to prayer; come to prosperity, come to prosperity; Allah is the Greatest, Allah is the Greatest; there is none worthy of worship except Allah).’ Then he called me when I had finished saying the Adhan, and he gave me a bundle in which there was some silver. I said: ‘O Messenger of Allah, let me be the one doing the Adhan in Makkah.’ He said: ‘I command you to do so.’ Then I came to ‘Attab bin Asid who was the governor of the Messenger of Allah in Makkah, and I called the Adhan for prayer with him upon the orders of the Messenger of Allah (ﷺ).”

[Saheeh Nasaai no. 631]

The Scholars took the opinion that Tarjee’ in Adhan is allowed. One has a choice between doing it or leaving it off.

Imam an-Nawawi rahimahullah said: “A group of the Scholars of Hadith and other than them took the opinion of having choice between performing Tarjee’ or leaving it off”[Sharh Saheeh Muslim (4/81)]

This is the opinion of Shaikul Islam Ibn Taymiyyyah [al-Fatawa al-Kubra (2/42)], Imam ash-Shawkaani [Sayl al-Jarar pg. 125], Shaikh al-Albani [Thamr al-Mustadaab pg. 208],  and Shaikh Ibn ul-‘Uthaymeen [Ta’leeqaat Ibn Uthaymeen ‘ala al-Kaafi]

And Allah knows best

Faisal Ibn Abdul Qaadir Ibn Hassan
Abu Sulaymaan

For the Iqamah there are two ways of doing it as Shaikh al-Albaani rahimahullah mentioned in Thamr al-Mustataab pg. 206. The first way is to pronounce everything twice as found in  the Hadith of Abu Mahthoorah radiallah anhu which I mentioned in my last post. The second way is as found in the Hadith of Anas

عَنْ سِمَاكِ بْنِ عَطِيَّةَ، عَنْ أَيُّوبَ، عَنْ أَبِي قِلاَبَةَ، عَنْ أَنَسٍ، قَالَ أُمِرَ بِلاَلٌ أَنْ يَشْفَعَ، الأَذَانَ وَأَنْ يُوتِرَ الإِقَامَةَ إِلاَّ الإِقَامَةَ‏.‏

Bilal was ordered to repeat the wording of the Adhan for prayers twice, and to pronounce the wording of the Iqama once except “Qad-qamat-is-salat”

[Bukhari no. 605]

Both are Sunnah and both allowed.

And Allah knows best

Faisal Ibn Abdul Qaadir Ibn Hassan
Abu Sulaymaan