All perfect praise be to Allah, the Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad, sallallahu ‘alayhi wa sallam, is His slave and Messenger.
The hadeeth that says that the prayer after using the Siwaak/Miswaak is better than the one without using it is not authentic, as we highlighted in fatwa 85904.
It is prescribed to use the Siwaak before every prayer. Abu Hurayrah, may Allah be pleased with him, narrated that the Prophet, sallallahu ‘alayhi wa sallam, said, "Were it not that I would be overburdening my community (or the people), I would have ordered them to use the Siwaak with every prayer." [Al-Bukhaari and Muslim]
Imaam An-Nawawi said, "If a person wants to perform a prayer concluded with Tasleem (the Salaam greeting) such as the Taraweeh prayer, Dhuha, four-unit-Sunnah-prayer of the Thuhr or ʻAsr, Tahajjud prayers, and the like, it is recommended to use the Siwaak before each two units of prayer because the Prophet, sallallahu ‘alayhi wa sallam, said, 'I would have ordered them to use the Siwaak before every prayer or with every prayer.'" [Al-Majmooʻ]
If you used the Siwaak for the Sunnah of the Fajr prayer, it is prescribed to use it again before the obligatory Fajr prayer. This applies to every prayer according to the wording of the hadeeth.
Allah knows best.