The veil of which Paul speaks in 2 Cor 3:14 is the veil that Moses put on his face to hide from the Jews the fact that his glory was departing:
2Co 3:13 and are not as Moses, who was putting a vail upon his own
face, for the sons of Israel not stedfastly to look to the end of
that which is being made useless,
Paul employs the fading glory of Moses' face to illustrate the fading significance of the Sinai covenant:
2Co 3:14 but their minds were hardened, for unto this day the same
vail at the reading of the Old Covenant doth remain unwithdrawn—which
in Christ is being made useless— 2Co 3:15 but till to-day, when
Moses is read, a vail upon their heart doth lie,
He is saying that God has hid from the Jews the obsolescence of the law, made obsolete by the arrival of the gospel:
2Co 3:7 and if the ministration of the death, in letters, engraved in
stones, came in glory, so that the sons of Israel were not able to
look stedfastly to the face of Moses, because of the glory of his
face—which was being made useless, 2Co 3:8 how shall the
ministration of the Spirit not be more in glory? 2Co 3:9 for if the
ministration of the condemnation is glory, much more doth the
ministration of the righteousness abound in glory; 2Co 3:10 for also
even that which hath been glorious, hath not been glorious—in this
respect, because of the superior glory; 2Co 3:11 for if that which
is being made useless is through glory, much more that which is
remaining is in glory.
One can easily relate Paul's argument to how bright a flashlight appears in a dark room but how dim it appears before the noonday sun. So when a Jew turns to the Lord the veil is "taken away" and they see that Moses no longer has any glory nor does the Sinai covenant:
2Co 3:16 and whenever they may turn unto the Lord, the vail is taken
Jeremiah had predicted that God would give the Jews a new covenant that was unlike the Sinai covenant in that it would be written upon their hearts and did not require teachers to explain it one to the other:
Jer 31:33 For this is the covenant that I make, With the house of Israel, after those days, An affirmation of Jehovah, I have given My law in their inward part, And on their heart I do write it, And I have been to them for God, And they are to me for a people. Jer 31:34 And they do not teach any more Each his neighbour, and each his brother, Saying, Know ye Jehovah, For they all know Me, from their least unto their greatest, An affirmation of Jehovah; For I pardon their iniquity, And of their sin I make mention no more.
Paul is dismissing any charge that he lacks credentials by describing his living and life-giving credentials written by the spirit in the hearts of men:
2Co 3:6 who also made us sufficient to be ministrants of a new covenant, not of letter, but of spirit; for the letter doth kill, and the spirit doth make alive.
When did/does the new covenant take effect?
"To the Hebrews" explicitly explains that the new covenant could not be in effect prior to the death of Jesus. Jesus was the "animal" who's death "sealed the deal" between God and the houses of Israel and Judah:
Heb 9:16 for where a covenant is , the death of the covenant-victim to come in is necessary, Heb 9:17 for a covenant over dead victims is stedfast, since it is no force at all when the covenant-victim liveth, Heb 9:18 whence not even the first apart from blood hath been initiated, Heb 9:19 for every command having been spoken, according to law, by Moses, to all the people, having taken the blood of the calves and goats, with water, and scarlet wool, and hyssop, he both the book itself and all the people did sprinkle, Heb 9:20 saying, 'This is the blood of the covenant that God enjoined unto you,'
This indicates that the chapters of the gospels prior to Jesus' death (and perhaps ascension, since Hebrews speaks of him cleansing the heavenly temple with his blood:
Hebrews 9: 18...This is why even the first covenant was not put into
effect without blood. 19When Moses had proclaimed every command of the
law to all the people, he took the blood of calves, together with
water, scarlet wool and branches of hyssop, and sprinkled the scroll
and all the people. 20He said, “This is the blood of the covenant,
which God has commanded you to keep.”e 21In the same way, he sprinkled
with the blood both the tabernacle and everything used in its
ceremonies. 22In fact, the law requires that nearly everything be
cleansed with blood, and without the shedding of blood there is no
23It was necessary, then, for the copies of the heavenly things to be
purified with these sacrifices, but the heavenly things themselves
with better sacrifices than these. 24For Christ did not enter a
sanctuary made with human hands that was only a copy of the true one;
he entered heaven itself, now to appear for us in God’s presence.
25Nor did he enter heaven to offer himself again and again, the way
the high priest enters the Most Holy Place every year with blood that
is not his own. 26Otherwise Christ would have had to suffer many times
since the creation of the world. But he has appeared once for all at
the culmination of the ages to do away with sin by the sacrifice of
Believing Jews were taught to celebrate the ratification of the new covenant during the celebration of the Passover:
1Co_11:25 In like manner also the cup after the supping, saying,
'This cup is the new covenant in my blood; this do ye, as often as ye
may drink it —to the remembrance of me;'
But to return to the original question, no, the reference to reading the old covenant referred to the reading of the writings of Moses and in particular the terms of the Sinai covenant.
YLT unless otherwise noted.