If some parts of extrabiblical (non-Old Testament or non-gospel) traditions can be cited as true in the New Testament, then it stands to reason and is quite plausible that other parts can be true (and hence, authoritative) without being cited in the New Testament.
Protestants simply assume without argument that anything that is fully authoritative must be in the Bible. But then why do Jesus, Matthew, Peter, James, Paul, and Jude cite traditions that we can’t find in our Bibles?
Our Lord Jesus Christ and the gospel writer St. Matthew did not believe in sola Scriptura. “Moses’ Seat” (Matthew 23:1-3) is an example of a tradition that is not in the Old Testament and yet confirmed as authoritative by Jesus and Matthew:
Then said Jesus to the crowds and to his disciples, “The scribes and the Pharisees sit on Moses’ seat; so practice and observe whatever they tell you, but not what they do; for they preach, but do not practice….”
Jesus says that the Golden Rule can be found in the law and the prophets, but it’s not in the Old Testament (least not in this positive form):
So whatever you wish that men would do to you, do so to them; for this is the law and the prophets. (Mt 7:12)
Matthew refers to another prophecy that cannot be found in the Old Testament:
And he went and dwelt in a city called Nazareth, that what was spoken by the prophets might be fulfilled, “He shall be called a Nazarene.” (Mt 2:23)
St. Peter did not believe in Sola Scriptura either. He also cites an oral, unbiblical tradition about Jesus that can’t be found in the written gospels:
For Christ also died for sins once for all, the righteous for the unrighteous, that he might bring us to God, being put to death in the flesh but made alive in the spirit; in which he went and preached to the spirits in prison, who formerly did not obey, when God’s patience waited in the days of Noah, during the building of the ark, in which a few, that is, eight persons, were saved through water. (1 Pt 3:18-20, emphasis mine)
St. Paul did not believe in sola Scriptura either. He cites the unbiblical tradition of a rock that follows Moses and the Israelites in the desert:
“For they drank from the supernatural Rock which followed them, and the Rock was Christ.” (1 Cor 10:4)
The Torah speaks only about a rock from which water issued, but rabbinic tradition amplified this into a spring that followed the Israelites throughout their migration.
Paul also names Pharaoh’s magicians even though the Old Testament gives them no names:
As Jannes and Jambres opposed Moses, so these men also oppose the truth, men of corrupt mind and counterfeit faith; but they will not get very far, for their folly will be plain to all, as was that of those two men. (1 Tim 3:8-9)
Paul also quotes a saying of Jesus that is not in the gospels:
…remembering the words of the Lord Jesus, how he said, ‘It is more blessed to give than to receive.’”
St. James also did not believe in sola Scriptura. He refers to a tradition about Elijah and a lack of rain that cannot be found in the relevant Old Testament passage 1 Kgs 17:
Eli’jah was a man of like nature with ourselves and he prayed fervently that it might not rain, and for three years and six months it did not rain on the earth. Then he prayed again and the heaven gave rain, and the earth brought forth its fruit. (James 5:17-18)
Jude also refers to an extrabiblical, tradition that cannot be found in the Old Testament:
But when the archangel Michael, contending with the devil, disputed about the body of Moses, he did not presume to pronounce a reviling judgment upon him, but said, “The Lord rebuke you.” (Jude 9)
Jude also directly quotes 1 Enoch 1:9, which is not in the Bible:
It was of these also that Enoch in the seventh generation from Adam prophesied, saying, “Behold, the Lord came with his holy myriads, to execute judgment on all, and to convict all the ungodly of all their deeds of ungodliness which they have committed in such an ungodly way, and of all the harsh things which ungodly sinners have spoken against him.” (Jude 14-15)
If Jesus, Matthew, Peter, James, Paul, and Jude were not Scripture Alone Christians, then why should anybody be?