Half banana, you wrote . . .
Bro Jeramy, just to clarify ---when Paul wrote "All scripture is inspired..." he was not referring to the Bible for it did not exist then.
For starters, note that I had very specifically stated "we also know that first century Christian writers of the Greek Scriptures, such as Peter and Paul, considered the Hebrew Scriptures
to be inspired." I did not say "the Bible."
We know for certain that the Hebrew Scriptures as a defined collection -- the Jewish bible, if you will -- had already existed for a number of centuries by the time the likes of Paul and others began to write in the first century. We know this collection as the Septuagint, the Hebrew Scriptures rendered into Koine Greek. And it has long been known that nearly all Hebrew Scripture quotations found in the Greek Scriptures are from this Septuagint.
We can confidently surmise that Paul, at 2 Timothy 3:16, was referring to the Hebrew Scriptures, particularly considering in verse 14 he refers to the "holy writings" (or "sacred writings" as other Bible translations/versions put it). It's logical to conclude that the one collection of writings he would certainly refer to as "holy" or "sacred" would be the Hebrew Scriptures.
Now, there is an interesting footnote to add to all this. It is very possible that, at 2 Timothy 3:16, Paul could have been referring to not only the Hebrew Scriptures but also to the Gospel of Luke, and maybe even Matthew and perhaps Mark as well. With this in mind, take a look at 1 Timothy 5:18:
For the scripture says, "You must not muzzle a bull when it is threshing out the grain," also, "The worker is worthy of his wages."
The first part of that passage, about not muzzling a bull, is a direct quote from Deuteronomy 25:4. The second part of the passage, which is of key interest here, is a direct quote from Luke 10:7: "The worker is worthy of his wages."
This is very highly suggestive that Paul was acquainted with the Gospel of Luke, which further suggests that by the time Paul wrote 2 Timothy, Luke (and very possibly Matthew, and perhaps even Mark) may have already been in circulation in the early Christian congregations. This lends credence to the conclusion of many scholars that Luke was written somewhere around 60CE (before Paul wrote his two letters to Timothy), perhaps earlier.
And so considering Paul refers to both sources of his quotes (i.e., Deuteronomy from the Hebrew Scriptures, and the Gospel of Luke) as "scripture," we can logically conclude that he held them on the same level in terms of sacred authority, if not divine inspiration.
In this way, when Paul writes "all Scripture is inspired of God" at 2 Timothy 3:16, we can conclude that he was clearly referring to the Hebrew Scriptures, and might very well have been referring to, at the very least, the Gospel of Luke, which is one of the four foundational books of the Christian Greek Scriptures.