Is it referring to "the sin" or to "the sin offering"?
NIV Hebrews 9:26 Otherwise Christ would have had to suffer many times since the creation of the world. But he has appeared once for all at the culmination of the ages to do away with sin by the sacrifice of himself.
The reason I suspect it refers to a sin offering as opposed to sin itself is the context and because such a reading is supported in Leviticus 5:9 LXX.
NIV Hebrews 9: 1Now the first covenant had regulations for worship and also an earthly sanctuary. 2A tabernacle was set up. In its first room were the lampstand and the table with its consecrated bread; this was called the Holy Place. 3Behind the second curtain was a room called the Most Holy Place, 4which had the golden altar of incense and the gold-covered ark of the covenant. This ark contained the gold jar of manna, Aaron’s staff that had budded, and the stone tablets of the covenant. 5Above the ark were the cherubim of the Glory, overshadowing the atonement cover. But we cannot discuss these things in detail now.
6When everything had been arranged like this, the priests entered regularly into the outer room to carry on their ministry. 7But only the high priest entered the inner room, and that only once a year, and never without blood, which he offered for himself and for the sins the people had committed in ignorance. 8The Holy Spirit was showing by this that the way into the Most Holy Place had not yet been disclosed as long as the first tabernacle was still functioning. 9This is an illustration for the present time, indicating that the gifts and sacrifices being offered were not able to clear the conscience of the worshiper. 10They are only a matter of food and drink and various ceremonial washings—external regulations applying until the time of the new order...
New International Version Leviticus 5:9 and is to splash some of the blood of the sin offering against the side of the altar; the rest of the blood must be drained out at the base of the altar. It is a sin offering.
9 καὶ ῥανεῖ ἀπὸ τοῦ αἵματος τοῦ περὶ τῆς ἁμαρτίας ἐπὶ τὸν τοῖχον τοῦ θυσιαστηρίου, τὸ δὲ κατάλοιπον τοῦ αἵματος καταστραγγιεῖ ἐπὶ τὴν βάσιν τοῦ θυσιαστηρίου· ἁμαρτίας γάρ ἐστιν·
Swete, H. B. (1909). The Old Testament in Greek: According to the Septuagint (Le 5:8–9). Cambridge, UK: Cambridge University Press.
I also notice that when speaking about sin, unlike Paul, the author of To the Hebrews uses the plural, "sins":
NIV Hebrews 9:7But only the high priest entered the inner room, and that only once a year, and never without blood, which he offered for himself and for the sins the people had committed in ignorance. ... 15For this reason Christ is the mediator of a new covenant, that those who are called may receive the promised eternal inheritance—now that he has died as a ransom to set them free from the sins committed under the first covenant. ... 28so Christ was sacrificed once to take away the sins of many; and he will appear a second time, not to bear sin[s], but to bring salvation to those who are waiting for him. ... NIV Hebrews 10: 2Otherwise, would they not have stopped being offered? For the worshipers would have been cleansed once for all, and would no longer have felt guilty for their sins. 3But those sacrifices are an annual reminder of sins. 4It is impossible for the blood of bulls and goats to take away sins. ... 11Day after day every priest stands and performs his religious duties; again and again he offers the same sacrifices, which can never take away sins. 12But when this priest had offered for all time one sacrifice for sins, he sat down at the right hand of God, ... 17Then he adds:
“Their sins and lawless acts I will remember no more.” 18And where these have been forgiven, [offering a] sacrifice for sin is no longer necessary. ... 26If we deliberately keep on sinning after we have received the knowledge of the truth, no sacrifice for sins is left,