The teachings of the Scripture of the time of Jesus (the Torah) was the only Scripture that they had then. All other teachings, by Jesus and by His disciples, were orally transmitted. The saints of the Church hadn't even started to write things down, compiling the Gospel accounts as we know it, until years after Jesus' death, when they realized that He wasn't returning within their lifetimes. And, before Jesus' time, back in the days of the Prophets, the Law was passed down orally, before ever getting written down by someone later on. In this way, adhering to a non-Sola Scriptura view (which actually predates Sola Scriptura, historically), is how Christianity (and Judaism before it) always operated.
You could say that in this way, there is a Biblical basis for operating non-Sola Scriptura, because the Bible clearly describes the faith being passed down in that manner. And if you believe that Jesus lives on through the Church, just as the Jews believed in the prophets, then I think it's pretty rational to do so in this manner. If you try and look, there isn't really a hard line or date where you can say "OK, from here on out, no new insights can be gathered--all that could ever be True has been written, and there is nothing more for us to figure out". In fact, I believe that the Church will forever be learning and growing deeper in understanding of God. He has been slowly revealing Himself more and more throughout the millenia, and so there is no reason to think that God stopped at some point.
In short, I guess you could say that it's traditional to not be Sola Scriptura, and that might sound like an odd reason at first glance, but I think it's precisely the point. The Church traditionally was not Sola Scriptura for centuries, and so that's why it might continue to this day.