Is it necessary to maintain the order you started in reading the Qur'an during a supererogatory (Sunnah) prayer when transitioning to an obligatory (Fard) prayer?

Reciting a chapter (Surah) after Al-Fatihah is Sunnah (recommended), not obligatory. What is obligatory is only reciting Al-Fatihah in every Rakat. Furthermore, adhering to the order of chapters is not obligatory. The Prophet (peace be upon him) did otherwise by reciting Al-Baqarah, then Al-Nisa, then Al-Imran in a single Rakat. It is also not prescribed to adhere to the order of chapters between different prayers.