The Hebrew word for "atonement" (kippur) literally means "a covering" but in the context of the Levitical priesthood and sacrifices it has to do with ritual cleansing. The root word, kopher, means "a ransom". The "kippur"/atonement offerings are literally the ransom or price of ritual cleansing... and that ransom involves blood.
In Leviticus 14:52 the blood of a bird is used to cleanse a house with "leprosy":
Leviticus 16 speaks of the high priest’s duty on Yom Kippur (the Day of Atonement) and the cleansing due to impurities:
Note that the atonement is primarily personal ("that he may make atonement for himself") then familial ("and for his household") and finally national ("and for all the assembly of Israel").
Ezekiel 43 speaks of the future time when the presence of G-d returns to the Temple (Ezekiel 43:1-2) and again commands an offering for cleansing:
The apostolic affirmation of this "atonement = cleansing" concept is found in Hebrews chapter 9:
The author of Hebrews affirms that the blood of goats and bulls (and the ashes of the red heifer) cleanses the flesh and then uses that truth in a lesser-to-greater (Latin: a fortiori , Hebrew: kal va-chomer) argument to prove "how much more" the blood of Christ cleanses our conscience.
Since the former is true then the latter is much more!
One final thought: although the Hebrew words for atoning or atonement are used over 100 times in the Old Testament, the words "atone", "atoning", or "atonement" are never found in the NASB translation of the New Testament. The Greek word used in the Septuagint for "atoning" (hilaskomai) is only used twice: by the publican in Luke 18:13 and in Hebrews 2:17 where it is (inaccurately in my opinion) translated into English as "propitiation" (an appeasement).
Scripture never says that Messiah is our atonement. It says He made atonement... with His own blood.
Blessed be His precious Name!