Yes, but none of it was widely available to geographically separated disciples and it wasn't part of "The Bible" until the Councils of Rome, Hippo, and Carthage put the 27 books of the New Testament together in 382 AD, 393 AD, and 397 AD.
There are some instances of Sacred Tradition in the Bible that are interesting. For instance, in Acts 20:35, Paul says the following:
"In all things I have shown you that by so toiling one must help the weak, remembering the words of the Lord Jesus, how he said, `It is more blessed to give than to receive.'"
These words are not recorded anywhere else in the Bible, including the 4 gospels, so this is one example of an oral teaching of Jesus being handed on to Paul,who hands it down to us.
Another example of this is in the book of Jude 1:9, which says the following:
"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."
This dispute, between the Archangel Michael and the devil over Moses' body, is nowhere to be found in the written text of the Old Testament. The books of Matthew, Hebrews, Timothy and Corinthians, all contain texts not found in the Old Testament. Sacred Tradition is the oral teaching of Jesus Christ handed on to the Apostles and the Church, which carries equal weight with the Church's book, the Bible.
You present argument of assertion, which itself becomes distortion. The "none of it...widely available" is absolute balderdash. There may have been some lesser, smaller works not included in collections of all, or were not uniformly seen as being equal to the Synoptic Gospels, Acts, and the greater portion of Paul's letters, but the remainder was broadly enough discussed or mentioned by various individuals prior to the time-frame you present, and looked upon as authentic from the time of Origin, with himself listing that which was considered Apostolic previous to his own time...
The works I mentioned, the greater bulk, was much MORE than less, uniformly known of, thus it can be assumed was present, in various churches East & West, long before the time frame you present. There could be minor exceptions, but that would leave those exceptions being argued as rule, while also still attempting to get away with the hinted at claim, that the books themselves were transmitted orally. Reading from them publicly, or in church assembly, is not the oral transmission which is being argued here.
It is not as if the works were first presented late in the fourth century, and previously transmitted only or chiefly orally, as you originally suggested (just not in those exact words).
It's more like there is record of them being cataloged at that time (late in the fourth century) by an organized group, organized with a purpose, rather than reliance upon their own church/group "tradition", and the more isolated written listings having cropped up previous. That does NOT mean those books were not previously regarded as authentic due to their Apostolic origins, but does mean that such was being written about in 382 AD, as that which had for centuries previous, by tradition, been regarded as authentic. Big difference. Can you see it? It's all about WHERE those books came from, not WHO it was that recognized those basic facts!
Now if one wanted to argue that the authenticity of that which eventually became more formally NT canon, was previous to the end of the fourth century, presented "orally", transmitted much by tradition, that would seem quite safe to assume to a great extent be true, even though there may have been some written listings scattered far and wide, that never made it into the historical record by having been mentioned and/or duplicated, much less surviving extant. What should be logical enough, is that prior to the first attempts to formally, as an assembly of churches/groups define NT canon, there had to have been a pre-existing sense of what that should be composed of.
For reason those books (which are now NT) were themselves widely accepted previous to that time, is the driving reason why those particular books and none other, were at that time cataloged. That such arose as hedge against gnosticisms, would be beside the point, for here we are still considering origins and faithful transmission of that which had been received.
As to disputations, to settle those matters, what can be seen to have been most widely applicable, but to fall upon reliance of that which was Apostolic in origin, setting aside all that which came afterwards? Though there is evidence of earlier isolated inclusions of a variety of NT apocrypha, the widest sense from as can be best reconstructed from earliest mentions, hold fast to the Apostolic, with significant unevenness in regards to NT apocrypha.