The context of the passage rests in the theme of the fullness of the sacrifice of Christ and the uselessness of animals and burnt offerings. Heb. 10 states, as a commentary on OT passages, that God takes no pleasure in those sacrifices but that He removed that and the One who came to do His will (vis a vis a perfect living fulfillment of the law and a perfect sacrifice of His own flesh) is established and by the doing of God's will the One who offered Himself now sanctifies us "once for all" as opposed to "yearly (on the Day of Atonement) for all (Israel)". Now, by the Holy Spirit, chapter 10 continues, the law is written in our hearts.
The altar of Christ is the Most Holy Place in heaven. Hebrews 9:24, 25. The priest is Christ Himself. And, as on the Day of Atonement, the priests could not partake of any of the sacrifice personally, but what was left was burned outside the camp, Jesus was so taken outside the camp. Chapter 11 says, in a nutshell, have faith. Chapter 12 starts with the admonition to live in faith as those faithful forebears listed in chapter 11 did....not with the hope of a future revelation of a better priest and better sacrifice, but considering Jesus, "the author and finisher of our faith", who has already come and IS that fulfillment.
Pay attention to those early verses in ch. 12 that explain how Jesus endured the cross, despising the shame, because of the joy that was set before Him. Hearken back to Philippians 2:2-10 and Philippians 4, how Paul there "does the math" and declares that this world and its ways (which by the time whenever Hebrews was written would have included the now defunct temple sacrifices) are "dung" and that dying with Christ daily is far better than all this life and this world and this order of things may offer. Why? Because, as Paul said in Philippians 2, every knee shall bow and every tongue confess that Jesus Christ is Lord to the glory of God the Father. Because the glory of God is better than the passing glories of here. Cf. 1 Cor. 15.
Heb 12 says, then, that the true worship is not an earthly place that we can touch, but to the city of God, the heavenly Jerusalem. We see this same Jerusalem coming down from heaven in Revelation. There is a real promise wrapped securely in God's hand and in His purpose to have for Himself what He created. (Rev. 4:11). Chapter 12 finishes with the thought of then not refusing Him, but serving Him with awe and reverence and fear.
Ch 13 begins with the continuing thought of living out this fear and this life based on the completed work of Jesus and the promises He still holds for His people: "let brotherly love continue". This whole way of life in the opening verses of ch. 13 are as superior to the Judaist way as the sacrifice of Jesus Himself is to the bulls and goats of the OT Law. And, as the priests could not consume the offering on the Day of Atonement, but placed it outside the camp (v. 10) , the Hebrews (and we as well) leave behind any thought of our offering as efficacious for any reason. We can not be satisfied from our offering, we cannot take any fulfillment for ourselves from His. He alone, for the glory of God alone, fulfilled God's desires for us in Himself which we will know in the city to come (v. 14.)
