Most of Catholic social teaching is part of the Ordinary Magisterium but finds its roots in Extraordinary magisterial teaching. It can be infallible, but as it is presented now, it is debated on what is infallible and what is not. It is easy to determine infallibility when it comes from an ecumenical council or when the Pope speaks ex cathedra. There is, however, one other source of infallible statements which make papal statements on matters of faith and morals outside of the ex cathedra scenario which make it very difficult to determine infallibility in the day to day.
If the Pope speaks in conjunction with the college of bishops, then it is ordinary magisterium. If the Pope speaks on behalf of the college of bishops as its head, then it is extraordinary magisterium.
If the Pope expresses his own personal opinion on faith and morals and how to apply them pastorally, then it is not infallible. If, however, the Pope says the exact same thing intending it to draw its authority from the office of the Papacy in the sense that he is the head of the universal college of bishops, then that same statement is infallible. This third source of infallible statements is what throws most moral theologians for a loop with regard to papal statements.
For example, did Pope Francis add the new paragraph on capital punishment into the Catechism using his authority as head of the college of bishops or did he add the paragraph using his ordinary magisterial authority as head of the Roman Curia? That’s the big question.
We also find the question of how far does infallibility extend. As infallibility is an exercise of the universal extraordinary magisterium, does it only apply to universal matters of faith and morals or can infallible statements be made in judgement of specific cases? Can the Pope infallibly say that something is immoral in a certain circumstance for a certain person or can it only be applied to universal statements on faith and morals since it is a function of the universal authority of the extraordinary magisterium?
Many moral theologians kind of just shrug and try to do the best they can.