In discussing the Day of Atonement it is helpful to accurately define the word "Atonement." Atonement does not mean forgiveness of sin. It is from the Hebrew word kaphar which means, "Covering". This is the one day a year when the High Priest is to create a covering for Israel, separating the sin from the sinner forever. This covering, is the robe of righteousness, that fine linen you have been looking for. This is the wedding day when Yeshua will marry His bride.
The sense in which the word is used is as a covering-protection from danger. The idea is that God is dangerous, as a consuming fire, sin does burn up, melt like wax. If common, mortal, finite and sinful man is to enter the presence of the Holy, Immortal, Infinite and Righteous God, the man must be covered (i.e. atoned for) or he will be consumed and destroyed by the presence of God.
For example, Adam and Eve in the Garden of Eden enjoyed the presence of God. They walked with him daily in the garden. Then they sinned. Their eyes were opened and they immediately realized that they were naked. They immediately realized that they were which covering. Instinctively they began to try to sew fig leaves together to cover themselves because they knew God was coming, and they knew they could no longer withstand his presence. They could not enter his presence which covering. Their effort to cover themselves, however, was futile; so when they heard God's voice and knew that he was entering the garden, they hid themselves. They could not come face to face with him.
After judging them, the book of Genesis tells you, that God made garments of skin to cover them. Thus the story of the fall of man leaves you with your first glimpse of atonement. God kills an animal to cover Adam and Eve. It is the first record of death in the creation, and it is a sacrifice meant for a covering. According to tradition, Feast of Trumpets is the anniversary of their creation. Is it possible that 10 days later was the day they sinned?