On the contrary, Pre-mil does not ignore the rest of the New Testament. That's how Premillenniists get to Pre-millennialism.
Here are some of the things Amillennialists ignore:
In Genesis chapter 3, we read of how Satan appeared in the Garden of Eden to beguile mankind, and this took place even during the sabbath rest of God (which is in Christ, the Creator), and it took place while Adam and Eve were living forever after God had breathed life into Adam, and Adam had become a living soul.
Revelation 12:9 calls Satan "the great dragon" and "the old serpent called Devil, and Satan, who deceives the whole world."
Satan's deception of the nations is also mentioned in Revelation chapter 20:
2 And he laid hold on the dragon, that old serpent, who is the Devil and Satan, and bound him a thousand years.
3 And he cast him into the abyss and shut him up and set a seal on him, that he should deceive the nations no more until the thousand years should be fulfilled. And after that he must be loosed a little time.
In scripture Satan is called the god of this aion (Age) and prince of the power of the air who works in the sons of disobedience, who will give the beast and false prophet his seat, power and great authority. The saints are warned they should be weary of his wiles and are told to resist him, and to put on the full armour of God because we do not wrestle against flesh and blood (2 Corinthians 4:3-4; Ephesians 2:2; 1 Peter 5:8-9; Ephesians 6:11-12; Revelation 2:9-10 & Revelation 2:13; 1 Thessalonians 2:18; James 4:7).
Amillennialists ignore all the above facts and scriptures, which hardly speak of a devil whose already bound.
2 Corinthians 4:3-4 and Ephesians 2:2 tell us about Satan's influence over the societies of this world, during this Age.
1 Peter 5:8-9; Ephesians 6:11-12; Revelation 2:9-10 & Revelation 2:13; 1 Thessalonians 2:18; James 4:7. tell us about Satan's activities in the world during this Age.
In Revelation chapters 12 through 13 we read about:
1. The dragon being cast out and cast down to earth when the woman's child had been caught up to God and His throne; and
2. The dragon going to war against the woman who had given birth to the Messiah the moment he saw he had been cast down to earth; and
3. When "the earth helped the woman", the dragon then going to war with "the rest of the woman's seed, who keep the commandments of God and have the testimony of Jesus Christ" (for the rest of the Age), culminating in the beast and false prophet's war against the saints at the close of the Age (which is seen in Revelation 13:7 & Revelation 13:15).
There's no mention of Satan's binding there either, but Amillenniialists will invent one. I mean, "it's necessary to invent a reference to Satan's binding, so Revelation 12 is as good a place as any" (per Amillennialists).