Has the Church stated any advantages or reasoning or prompting to re-formulating positively the Catholic Church's salvation doctrine? The Church in Catechism of the Catholic Church: "Outside the Church there is no salvation", 846, has re-formulated positively "Outside the Church there is no salvation", often repeated by the Church Fathers to mean that all salvation comes from Christ the Head through the Church which is his Body.

Looking at St. Vincent of Lerins' description for what is truly and properly 'Catholic', in The "Vincentian Canon", AD 434 (3) "Now in the Catholic Church itself we take the greatest care to hold that which has been believed everywhere, always and by all.", it would appear that the Church would be very hesitant and reluctant to re-formulate any of its doctrines especially when it was often repeated by the Church Fathers.

The question is what has the Church stated as the advantages or reasoning or prompting to the re-formulation?
This re-formulation is nowhere stated to supersede the "Outside the Church..." formulation. It is a rephrasing which the Catechism uses to explain the original formula. An obvious advantage of having two reformulations is that it allows one to understand a single truth in multiple ways. Thus, there is the original formulation which you state, which certainly brings to mind the last sentence of paragraph 846, to which you link:


  Hence they could not be saved who, knowing that the Catholic Church was founded as necessary by God through Christ, would refuse either to enter it or to remain in it.


This is a re-translation of a quote from the Dogmatic Constitution Lumen Gentium:


  In explicit terms [Christ] Himself affirmed the necessity of faith and baptism and thereby affirmed also the necessity of the Church, for through baptism as through a door men enter the Church. Whosoever, therefore, knowing that the Catholic Church was made necessary by Christ, would refuse to enter or to remain in it, could not be saved.


This last sentence sounds somewhat similar to "No salvation outside the Church"; the difference of course is that it allows that people who are outside the Catholic Church, and who have no knowledge of it but haven't actually rejected it outright, might be saved. This is not necessarily incompatible with older statements; for example Pope Pius IX's 1863 encyclical Quanto Conficiamur Moerere reads, in part:


  There are, of course, those who are struggling with invincible ignorance about our most holy religion. Sincerely observing the natural law and its precepts inscribed by God on all hearts and ready to obey God, they live honest lives and are able to attain eternal life by the efficacious virtue of divine light and grace. Because God knows, searches and clearly understands the minds, hearts, thoughts, and nature of all, his supreme kindness and clemency do not permit anyone at all who is not guilty of deliberate sin to suffer eternal punishments.


The previous sentence to this quote reads:


  Eternal salvation cannot be obtained by those who oppose the authority and statements of the same Church and are stubbornly separated from the unity of the Church and also from the successor of Peter, the Roman Pontiff.


It appears, then, that Pius IX was aware of these two aspects of Extra Ecclesia Nulla Salus, and did not feel they were contradictory. Nor does the Church now feel that they are contradictory:


  it must be firmly believed that "the Church, a pilgrim now on earth, is necessary for salvation: the one Christ is the mediator and the way of salvation; he is present to us in his body which is the Church. He himself explicitly asserted the necessity of faith and baptism (cf. Mk 16:16; Jn 3:5), and thereby affirmed at the same time the necessity of the Church which men enter through baptism as through a door". This doctrine must not be set against the universal salvific will of God (cf. 1 Tim 2:4); "it is necessary to keep these two truths together, namely, the real possibility of salvation in Christ for all mankind and the necessity of the Church for this salvation"


Dominus Iesus (a declaration of the Congregation for the Doctrine of the Faith); the quotes cited are respectively from Lumen Gentium, a document of Vatican II, and Redemptoris Missio, an encyclical letter of Pope John Paul II

It's thus not truly clear whether this was completely a re-formulation, or simply a re-emphasis of a neglected part of Catholic teaching. There are no specifically published justifications that I can find for it; it seems to have been around for quite some time.