The New Testament is just as much 'scripture' as the Old Testament, and in some sense we believe the Old Testament is scripture because the New Testament records our Lord's full attestation of its authority as such. Inspired works are nothing in comparison. Even my post could be inspired by God as Christians often have inspired thoughts.
I think when people try to trace the overall idea of how scripture came to be believed as 'scripture' we forget the basic idea. God sent Prophets and, later on, Apostles to be his messengers, to deliver his word. Apostles were even greater 'messengers' of the oracles of God than Prophets. A greater covenant was under their instruction as infallible guides to the church. In the Old Testament, prophets, once established as credible, were not to be trusted blindly. Only but by testing the truthfulness of their words and prophecies as they continued to make them, would they be received. If they declared something as 'the word of the Lord' which was not true, they were to be stoned. (Deuteronomy 13:1-5)
There was great seriousness in someone saying 'Thus saith the Lord' or speaking as though they were bringing the 'very words of God' (and rightly so). Any misrepresentation was deadly. So we must start with this understanding that there was an idea of a Prophet, who was a messenger. He would speak in an authoritative manner as though his words were God's (because they were) and God gave such a person unusual certainty and assurance with these words, removing all doubt. Not that every word they spoke (like 'pass the butter please') was God's word, but every word they delivered to the church as a guide and message was of God.
At the highest level:
The scriptures alone have the authority to declare themselves to be so. No church council can determine what is scripture, they can only admit after careful investigation what is self evidently declared within the scriptures themselves. These are tested and confessed to having the authoritative and infallible revelation of God's mind with no contrary evidence to maintain reasonable dispute.
No matter what we canonize, the scripture will insert itself into our view and demand our obedience as it truly is the word of the Lord. God will not leave us without all that is necessary for life and salvation. I have read portions of letters claiming to own a right into the Protestant Bible, but have never sensed the pure authority of God in them. In fact it really puzzles me how many people fail to see the poor attitudes and doctrinal imperfections in so many 'inspired' books that are proposed as additions to our canon.
Now for those books that the church has received (I am speaking from a Protestant view and canon) a few things had to occur for the church to receive them as scripture. First, somebody claimed to be a Prophet or an Apostle, and their words were recorded by themselves or by others. Second, these words were sent out with an air of extreme importance and ultimate infallible authority. There are not many candidates meeting the first two criteria. Third, fake ones were tossed. They did not have the divine purity, power and truth in them as attested by the conscience of the church and which could be proven to others by contradicting something in scriptures already received. The three progressions occurred for all books in our Bible, and no other writings, however 'inspired', are included due to their obvious imperfections or lack of claim to be prophetic or apostolic in origin.
The notion that the Old Testament is the only scripture is absolutely inconsistent with the notion of an Apostle. It is like saying Prophets are great and God would have us collect their words and call them scripture, but those messengers even greater than Apostles, no, even the words of the Lord Jesus Christ himself, they are just 'inspired'? Nothing could be more ludicrous and in opposition to the concept of an Apostle. All scripture is equally scripture but if we had to rank different books, the New Testament would be more so simply from the height of the authority of the messengers and the level of brilliance and progression of the revelation.
On a more technical nature, one can trace the assumption of the New Testament being 'obvious scripture' and expect a direct argument never to be made. There was no direct argument in the Old Testament to receive the words of a prophet as scripture canonized in a little book called the Bible. It was common sense that the words of a prophet came from God and also common sense that the words of an Apostle was from God. This is spiritual common sense which requires no proof. It is like asking for proof that there is a God. He just is and his word says so.
For example in Acts we see the gospel is just assumed to be God's word, not just inspiration. The gospel as preached by the Apostles was always considered 'the word of the Lord', when written this became scripture by default. All one has to do is search the phrase and many passages in Acts appear:
The next Sabbath almost the whole city gathered to hear the word of the Lord.
(Acts 13:44, NIV)
The word of the Lord is scripture as provided by Prophets or Apostles, or the Lord himself. Paul considered the message he delivered in his letters as 'the word of the Lord' also. (1 Thessalonians 4:15, Galatians 1:11). In fact everything he said was assumed to be the Lord's word so that if on a rare occasion he offered his own opinion, he clarified as much. (1 Corinthians 7:12).
Some New Testament books were not yet written by the time 1 Timothy was written, so that they were not referred to as the 'scriptures' in a final sense, as the Old Testament version was. (The Old Testament canon was not collected and 'canonized' as such but these were known to be complete by the Apostles as the New Testament was being written and still incomplete. Therefore they were not referred to as a set). However, even already in Timothy the 'Spirit expressly saying' is agreed by most commentators to be referring not to just the Hebrew scriptures but prophecies made by Jesus and others more recently in New Testament letters. (1 Timothy 4:1)
A fairly standard verse that people use on the technical side is 2 Peter 3:15-16:
And count the patience of our Lord as salvation, just as our beloved brother Paul also wrote to you according to the wisdom given him, as he does in all his letters when he speaks in them of these matters. There are some things in them that are hard to understand, which the ignorant and unstable twist to their own destruction, as they do the other Scriptures.
Clearly Peter 'assumes the writings of Paul are scripture' otherwise he would be guilty of gross miscommunication on a very important matter. Some think Peter is arguing directly that Paul's writings are scripture. I am in the camp of thinking he just assumes it as obvious due to his Apostleship and assumes the reader to understand it obvious as well.
Note: I take for granted that it is understood that an Apostle is greater than a Prophet. If one needed support for this argument I would present 1 Corinthians 12:28, the fact that the Lord himself called the twelve Apostles, the fact that the writings of the Apostolic age contain greater revelations than those of the Prophets and that the gospel is so much superior that every believer is considered greater than the greatest prophet, John the Baptist. His greatness only occurs from his nearness in time to the Messiah. (Matthew 11:11)
Inspired works are nothing in comparison to scripture as only scripture can test the truthfulness of their claimed inspiration. Scripture is the word of God the church has received from Prophets or Apostles. The 'greatest of these', if such a word can be used to compare scripture with scripture due to the weight of the revelation contained in them, is the New Testament.