Edersheim, who I consulted later, in the Temple and its sacrifices comments that if the practice had been in force during the time of the Septuagint we might have expected some clues in the LXX text.  If he is right then it might not have been during the second Temple era but that of Herod's temple.

Edersheim writes..."We may here at once state, that the later Jewish practice of pushing the goat over a rocky precipice was undoubtedly on innovation, in no wise sanctioned by the law of Moses, and not even introduced at the time the Septuagint translation was made, as its rendering of Leviticus 16:26 shows. "

the first goat is the picture of the punishment of sin, the second is the picture of the remission of sin.  In fact, this 'letting go' of sins is the foundation stone of the Bible concept of 'remission'.  Sins are remitted, sinners are forgiven.  Many translations obscure this distinction.

Christ, of course,'became sin' (2Cor 5). We have a sobering picture here of the Saviour as the 'goat' whose life is forfeit and the 'goat' carries away the sins.  Both goats fused together give the fuller shadow of the coming reality.

The later addition of 'pushing the goat off the cliff' confuses the separate idea of punishment and function.  The first goat speaks of the punishment, the second of the result of that punishment.  Human addition to divine instruction seldom makes the story clearer.