You write that on the Day of Atonement the high priest made three appearances in the Most Holy Place. Would his appearance with the blood of a bull not make this four appearances?
This text is from a letter written by David Gooding in 2012.
It is perfectly true, as you observe, that, on the Day of Atonement, the high priest had already been in with the blood of a bull (see Leviticus 16:11–14). In response to my article in The Perfect Saviour, IVP, 2012, p. 86, you ask if this should be four appearances.
But Leviticus 16:11–14 are instructions for what Aaron shall do with the bullock of the sin offering which is for himself: and he shall make atonement for himself and for his house, and shall kill the bullock of the sin offering which is for himself, and so shall enter in within the veil. The ancient high priest in Israel was a sinner, and therefore, before he could appear before God on behalf of the people, he had to bring the appropriate sin offerings for himself and for his house.
Of course, our Lord was not a sinner, and his sacrifice at Calvary was not for his own sins, and therefore the Epistle to the Hebrews distinguishes very carefully what the high priest in Israel did for himself and his own house, as distinct from what he did for the people.
My observation in the article, therefore, is deliberately limited to the account of what Aaron did on behalf of the people. I hope this explanation is sufficient to clear up the difficulty.