If the first reading is correct, and it should be translated as "They, like Adam, have transgressed the covenant," then the covenant could be the first six of the seven commandments of Noah.
The Rambam writes in Hil. Melakhim uMilkhamoth 9:1
Six precepts were commanded to Adam:
a) the prohibition against worship of false gods;
b) the prohibition against cursing God;
c) the prohibition against murder;
d) the prohibition against incest and adultery;
e) the prohibition against theft;
f) the command to establish laws and courts of justice.
Even though we have received all of these commands from Moses and, furthermore, they are concepts which intellect itself tends to accept,
it appears from the Torah's words that Adam was commanded concerning them.
He appears to be sourcing this from Sanhedrin 56b where two interpretations are provided. I've only pasted here the first, but you can read the rest of the daf here.
Whence do we know this? — R. Johanan answered: The Writ saith: And the Lord God commanded the man saying, of every tree of the garden thou mayest freely eat. And [He] commanded, refers to [the observance of] social laws, and thus it is written, For I know him, that he will command his children and his household after him, and they shall keep the way of the Lord, to do justice and judgment. The Lord — is [a prohibition against] blasphemy, and thus it is written, and he that blasphemeth the name of the Lord, he shall surely be put to death. God — is [an injunction against] idolatry, and thus it is written, Thou shalt have no other gods before Me. The man — refers to bloodshed [murder], and thus it is written, Whoso sheddeth man's blood, by man shall his blood be shed. Saying — refers to adultery, and thus it is written, They say, If a man put away his wife, and she go from him, and became another man's. Of every tree of the garden — but not of robbery. Thou mayest freely eat — but not flesh cut from a living animal.
Since the derivation in the Talmud is from the verses related to the prohibition of eating from the Tree of Knowledge in Eden, I assume that is not a separate covenant. Thus I'm not aware of any other covenant Adam took part in, so I'd assume it must be this one.