In addition to Susan's answer, I suggest "two hairy males from among the female goats" is an alternate meaning.
First, the Day of Atonement instructions in Leviticus 16 have the unusual aspect that none of the prescribed animals are required to be without blemish or of a certain age. While the sin and burnt offerings may logically be governed by the instructions elsewhere, the two atonement goats are unique to this ritual and Leviticus 16:5 is the selection criteria.
I believe the instructions would be considered in the practical terms of raising goats. Also, the most relevant Biblical context should be how the LORD God made and fulfilled His promises to Abram. That is, the people are going to understand שְׁנֵֽי־ שְׂעִירֵ֥י עִזִּ֖ים in everyday terms consistent with the historical context of the LORD God's work to make a nation of the children of Israel.
The first use of וְעֵ֥ז occurs when the LORD affirms His covenant with Abram:
And he said unto him, Take me an heifer of three years old, and a she goat [וְעֵ֥ז] of three years old, and a ram of three years old, and a turtledove, and a young pigeon. (Genesis 15:9 KJV)
It is next used in Rebekah's plot for Jacob to deceive Isaac into believing he is Esau:
Go now to the flock, and fetch me from thence two good kids of the goats [עִזִּ֖ים]; and I will make them savoury meat for thy father, such as he loveth: (Genesis 27:9 KJV)
And she put the skins of the kids of the goats [הָֽעִזִּ֔ים] upon his hands, and upon the smooth of his neck: (Genesis 27:16 KJV)
It then figures prominently in how Jacob cares for Laban's flocks:
I will pass through all thy flock to day, removing from thence all the speckled and spotted cattle, and all the brown cattle among the sheep, and the spotted and speckled among the goats [בָּעִזִּ֑ים]: and of such shall be my hire. (Genesis 30:32 KJV)
Also in Genesis 30:33, 35, and 31:38 where it is used in the context of breeding as Jacob builds his flocks from the newborns.
It is also used in describing the events of Jacob returning to meet Esau:
Two hundred she goats [עִזִּ֣ים], and twenty he goats [וּתְיָשִׁ֖ים], two hundred ewes, and twenty rams, (Genesis 32:14 KJV)
The first use of שָׂעִיר occurs when Jacob describes Esau to Rebekah's in response to her plan for him to take Esau's place:
And Jacob said to Rebekah his mother, Behold, Esau my brother is a hairy [שָׂעִ֔ר] man, and I am a smooth man: (Genesis 27:11 KJV)
The language in Leviticus 16:5 to describe the two goats chosen for the day of Atonement rite have connections both to the covenant with Abram and how Jacob and his descendants became set apart from Esau and his descendants.
The ending of שָׂעִיר (śāʿı̂r) denotes the masculine. So one possible literal meaning is: "two male goats from among the female goats." This might seem odd and may invoke objections on textual grounds; yet raising goats involves breeding which is done by keeping separate male and female flocks. Modern translations tend to obscure the practical reality that choosing a male goat normally means first choosing the herd. Should the goats come from exclusively male (תָּ֫יִשׁ) herds or from the males selectively placed in female (עֵז) herds? In other words, from the perspective of raising goats, Aaron is directed to choose from specific flocks. So while עֵז (ʿēz) can mean flocks in general, practically speaking Aaron is going to go to the עֵז not תָּ֫יִשׁ to get the his goats.
I believe the phrase "take two males from among the female flocks" is the correct selection criteria. Just as Jacob choose specific males to place in the female flocks to breed and kept separate male and female flocks when returning to Canaan, Aaron is to choose two males from among those that have been used for breeding.
Seeing the allusions of the Day of Atonement goats to the events of Jacob and Esau brings another reference to light. The instructions for handling the scapegoat follow Rebekah's and Isaac's instructions sending Jacob away:
And Isaac sent away (וַיִּשְׁלַ֤ח) Jacob... (Genesis 28:5 KJV)
But the goat, on which the lot fell to be the scapegoat, shall be presented alive before the Lord, to make an atonement with him, and to let him go (לְשַׁלַּ֥ח) for a scapegoat into the wilderness. (Leviticus 16:10 KJV)
Finally, the instructions include the "sent one" returning back to the people from he was sent:
And he that let go (וְהַֽמְשַׁלֵּ֤חַ) the goat for the scapegoat shall wash his clothes, and bathe his flesh in water, and afterward come into the camp. (Leviticus 16:26 KJV)
Given these connections to Jacob and Esau, adding "hairy" to the meaning is in keeping with the intent of the ritual which seems to be built upon the real events in the life of Jacob and so forming the nation promised to Abram.
I believe the better understanding is two hairy males from among the female goats.