(1) All human faculties and organs are directed towards the chief purpose of God’s design.
(2) All sexual organs are directed towards the purpose of procreation
(3) Any technological or willful interruption of this directed purpose is unnatural and therefore wrong in going against God’s design.
(4) Contraception is such an interruption of this natural purpose
(5) Therefore, contraception is morally wrong.
My best guess is that this weird argument is either a conflation of several different arguments or is just Hackett's own homebrew guess of what the whole argument must be based on fragments or brief summaries. Typically, there is not a single natural law argument for or against anything; there are multiple lines of argument. This is sometimes not made clear in particular discussions, but is obvious enough if you compare multiple arguments or look at any serious survey. Typically, natural law arguments are not based on "God's design", although conceivably one might just mean by this the ordering of nature to assistance in divine creation that does come up occasionally in particularly theological versions of arguments against contraception. Typically, the purposes involved in natural law arguments do not involve the purposes of organs but the purposes of actions, and are not concerned with biology but with rational of practical action using our biology. Typically, the purposes involved need to be specifically concerned with the common good of the entire human race, not just any purposes. Actually, this last one is not merely 'typically'. While the point is not always made explicit, an argument that does not have this element does not involve natural law theory: it is precisely the point of every paradigmatic version of natural law theory that principles of practical reason are natural law insofar as they concern common good. (One can generally re-formulate arguments against contraception, for instance, in terms of the having and raising of children as part of the good shared in common by the human race.)
Since Janet Smith seems to have occasioned Hackett's reflection, let's actually look at what Smith herself says about the arguments against contraception. In her book Humanae Vitae: A Generation Later, she gives six such arguments, summarized in syllogism form (pp. 98-99):
A1. What is artificial is unnatural and wrong.
A2. Contraception is artificial.
A3. Therefore, contraception is unnatural and wrong.
B1. It is wrong to interfere with the natural purposes of organs and acts.
B2. The purpose of sexual intercourse is reproduction (of the species).
B3. Therefore, since contraception interferes with the purpose of sexual intercourse, contraception is wrong.
C1. It is wrong to impede the procreative power of actions that are ordained by their nature to the generation of new human life.
C2. Contraception impedes the procreative power of actions that are ordained by their nature to the generation of new human life.
C3. Therefore, contraception is wrong.
D1. It is wrong to impede the procreative power of actions that are ordained by their nature to assist God in performing His creative act that brings forth a new human life.
D2. Contraception impedes the procreative power of actions that are ordained by their nature to assist God in performing His creative act that brings forth a new human life.
D3. Therefore, contraception is wrong.
E1. It is always wrong to have a contralife will.
E2. The use of contraception entails a contralife will.
E3. Therefore, contraception is wrong.
F1. It is wrong to destroy the power of human sexual intercourse to represent objectively the mutual, total self-giving of spouses.
F2. Contraception destroy the power of human sexual intercourse to represent objectively the mutual, total self-giving of spouses.
F3. Therefore, contraception is wrong.
Each of these, of course, is merely a summary for the purposes of classification; the meat of the argument in each case will be in the arguments for the major and the minor. (It also might be worth pointing out that, for similar reasons and despite the handiness of the syllogistic format for classification purposes, it sometimes obscures key elements that are usually work in arguments of that family. For instance, B-style arguments are usually specifically about the cooperative character of sexual reproduction, and F-style arguments are usually about meaning-suppression, although you wouldn't necessarily recognize that just from these coarse-grained headings.) Here we are only concerned with argument classification, so we will stay at this level. Smith herself rejects classes A, B, and E. A, she says, is generally given as the argument only by critics, and appears to muddle together several different conceptions of 'natural' in ways not consistent with natural law arguments in general. B, she says, you can find among a handful of theologians, and was discussed earlier in the twentieth century (although usually to express doubts about it); probably the version most worth taking seriously is that of Richard Connell. She notes that arguments against this class of argument are often more ad hoc than is usually admitted, but does not accept it herself, nor regard it as in any way a typical argument. E she thinks insufficiently precise to be adequate.
You'll notice that Hackett's version mixes and matches. His (1) is (interpreted most charitably) a D-type premise; his (2) is clearly a B-type premise; and his (3) seems to be an A-type premise. This is confirmed by the way in which he responds to the argument he gives. Since Hackett specifically makes his argument about the purpose of reproductive organs, like B, it's unsurprising that he spends a lot of time arguing that it can be rejected by accepting a "naturalized teleology of biology". Since Hackett specifically makes his argument depend on a theological principle about divine purposes, like D, it's unsurprising that he says, "Without God’s existence, the argument does not have legs to stand on." Since he uses an A-type premise, it is unsurprising that he appeals to human tool-making as a major part of his counter-argument to the 'typical' argument against contraception.
What Hackett seems to do, in other words, is muddle together a number of arguments that a natural law theorist like Smith regards as distinct, and says that this muddled mongrel is the way the argument 'typically' goes, and then puts other arguments in a 'theology of body' box. This type of confusion shows why careful classification of arguments is of serious importance to philosophical inquiry, despite the rather cavalier treatment it gets from philosophers. The natural history of arguments in the field needs to be done properly for philosophers to contribute things that are useful to a discussion.
Janet E. Smith, Humane Vitae: A Generation Later, The Catholic University of America Press (Washington, DC: 1991).