Is it permissible to supplicate aloud after the prayer, and what is the legitimate manner for doing so, knowing that Ibn Masoud used to know the end of the prayer in the era of the Messenger of Allah, peace and blessings be upon him, by hearing the supplication aloud?

The aforementioned Hadith is narrated by Ibn Abbas, not Ibn Mas'ud. Its purport is that raising one's voice with dhikr (remembrance of Allah) when people finished the prescribed prayer was [customary] during the time of the Prophet, peace and blessings be upon him, and Ibn Abbas used this as evidence for their departure. The Hadith is general in its mention of dhikr, and supplication is part of it.