This site has some insightful comments about this. Notice Mary as our Advocate has not yet being declared as a dogma, but as you indicated, it is stated in the Salve Regina, a one thousand year old prayer. First, notice that Jesus is the mediator between God the Father and Us:
God the Father dwells in unapproachable light. No human being has, or can see, the face of God (Jn 1:18) because of our impurity and sin. Nothing flawed could stand in the presence of God. Therefore, the Father, sent Jesus, to become a man and by His act of redemption, He reconciled man with God but also continually mediates between the people He has won and the Father.
We know this from Jesus himself (John 14:6):
Jesus said to him, “I am the way, and the truth, and the life; no one comes to the Father, but by me.
But to go to Jesus, the Church and NT show us that we can also have some "advocated" that pray for us, or through which we can be the object of prayer. The article above states:
It would certainly be more humble for us to have someone go to the Lord on our behalf. In fact, we do this all the time when we ask others to pray for us. Often, our prayers are answered more effectively this way because God desires for us to be humble and to depend on one another in prayer. This not only helps in our need but also builds up the body of Christ, the Church.
Who is best at this task than the Blessed Virgin Mary. From the very fact of her "Yes" to the angel, which enabled our salvation, to Jesus' own commandment to take St. Mary to "our home" (John 19:27), evidence of this is abundant (see the article for some).
It is important however to differentiate Mary as "official" mediator of graces and as advocate. The former is the belief of Mary Mediatrix, which also is not (yet?) a dogma. Here we are dealing with the latter. I quote the article above, in extenso:
However, at this point I had better make a distinction between the teaching of Mary being mediatrix of all graces and the devotion of choosing to go through her to Jesus and to the Father. Mary as Mediatrix or Dispensatrix of All Grace is something we believe and, as defined above, needs to be acknowledged as a truth and as such is a contemplative act, while the devotion to Mary as our mediator is an active practice of our spiritual life. This latter devotion to Mary as our mediator with the one Mediator, Jesus, comes under the title of Mary as Advocate. This word gives a further dimension to Mary as our mediator. She is not just interceding for us she is advocating, actively championing our cause. This process of Mary mediating and advocating for us is completely and correctly explained in St. Louis Marie De Montfort's treatise, "True Devotion to Mary" and is too large a topic to be dealt with in this short work. Nevertheless, for those who would know more about this practice, I recommend St. De Montfort's book which is reported to have transformed the life of the man who was to become Pope John Paul II.