The relation of Section II to Section I of the Groundwork is not obvious; and a question about it is quite in place. The topic is largely neglected.
Groundwork Part I
Section 1: Inducing the Desire for Philosophy
In purporting to show that the “supreme moral principle” is already
present in “popular moral consciousness” —and hence may be recovered
through philosophical analysis — section 1 of the Groundwork takes the reader through a series of related arguments. First, Kant claims that his readers already know that the only unconditionally good thing is a good will.
They know that the good will is an incomparably higher good than all the
ends we associate with happiness—“Power, riches, honor, even health”—
and all the virtues to which the pagan philosophers aspired: “Moderation
in affects and passions, self-control, and tranquil reflection” (AK, 4:393, 394;
PP, pp. 49, 50). Next, Kant moves to elucidate this still somewhat esoteric
conception of the good will by showing that it is already contained in the
popular idea of doing one’s duty for its own sake (see AK, 4:397–400; PP,
pp. 52–55). He then argues that his concept of duty must be understood as
the determination of the will through the mere idea or thought (Vorstellung)
of duty (see AK, 4:401–2; PP, pp. 56–57). Finally, Kant concludes that in
constructing this conception of the moral principle he has done nothing
more than clarify a principle already present in ordinary moral consciousness (see AK, 4:403–5; PP, pp. 58–60). (Ian Hunter, 'The Morals of Metaphysics: Kant’s Groundwork as Intellectual Paideia', Critical Inquiry , Vol. 28, No. 4 (Summer 2002), pp. 908-929: 914-5.)
...
Kant’s appeal to his students’ sense of “duty for its own
sake” — formed no doubt in religious, military, and pedagogical institutions
requiring unconditional obedience—is no simple elicitation of evidence for
the moral law’s preexistence. In fact it is a means by which his students can
be induced to subject themselves to the law as something that already commands them from within. The crucial thing to note in this regard is Kant’s
initial characterization of duty: “We shall therefore take up the concept of
duty, which contains that of a good will though under certain subjective
limitations and hindrances, which, however, far from concealing it and
making it unrecognisable, rather bring it out by contrast and make it shine
more brightly” (AK, 4:397; PP, p. 52). Here Kant provides his students with
a new way of relating to their (still unfocused) sense of duty. By treating its
compulsive character as arising from the form in which a pure rational will
encounters the “subjective limitations and hindrances” of their sensuous
natures, Kant incites his students to view their ordinary sense of duty (no
matter what its source) as if it were their dimly “sensed dependency” as
material beings on the self-governing community of intelligences in which
they participate as immaterial (rational) beings. (Hunter: 917.)
The 'form in which a pure rational will encounters the “subjective limitations and hindrances” of [our] sensuous natures' is that of a universal law to which as such we conform our actions.
Such a law is a categorical imperative and it is implicit in 'popular moral consciousness'. Section I thus draws out the idea of the categorical imperative from our ordinary moral thinking.
Groundwork Part II
Section II approaches the categorical imperative not from the side of ordinary moral thinking but from that of metaphysics or philosophical analysis. A purely rational will could only be a will that conforms its actions to universal law. Rationality and universality, the rational and the universally lawlike, are inseparably conjoined. And so Kant delivers the categorical imperative from a different angle - that of rationality as distinct from ordinary moral thinking.
He is also able later in Section II to formulate different but mutually entailing versions of the categorical imperative, a task not ventured in Section I.
Section 2: Teaching Transcendence
Having secured an audience disposed to view itself as the bearer of a pure
but latent moral law, in section 2 of the Groundwork Kant shows how this
law may be revealed, requiring his students to rise from “popular moral
philosophy to the metaphysics of morals.” This transition takes the form of
a series of arguments designed to “deduce” the moral law and to show the
necessity of metaphysics for obtaining this insight.
...
Kant ... presents these arguments as if they were solving a philosophical problem, namely, the problem of showing how a categorical imperative is possible. We can show the possibility of technical imperatives
(“imperatives of skill”)—the rules of geometry, for example—by demonstrating their analytic necessity for achieving a particular technical end, such
as the construction of a mathematical figure (AK, 4:417; PP, p. 70). Further,
we can show the possibility of prudential imperatives as the empirically necessary means to certain kinds of happiness; although here human disagreement over the ends of empirical happiness, and the uncertainty of their
worldly attainment, means that prudential imperatives lack the unified and
unconditional character of the moral law (see AK, 4:418–19; PP, pp. 70–71).
How though, asks Kant, can we show the possibility of the moral law’s categorical imperative, given that this is by definition unconditional, hence
independent of all empirical ends or goods capable of showing its necessity
as a means?
This is the problem, Kant argues, whose solution hinges on the transition
to metaphysics, which enables the philosopher to transcend the world of
empirical ends and means and to propose a “solely a priori” solution. Given his conception of metaphysics, as the discipline permitting access to a domain where thinking natures act independently of all external empirical
ends, Kant’s solution is to propose that the mere thought or concept of the
categorical command might itself reveal its propositional content—and to
this degree its possibility—independent of all need to relate this command
to some empirical object or end: “In this task we want first to inquire
whether the mere concept of a categorical imperative may not also provide
its formula, containing the only proposition that can be a categorical imperative” (AK, 4:420; PP, pp. 72–73). The only proposition that can be a
categorical imperative is, of course, “act only in accordance with that maxim
through which you can at the same time will that it become a universal
law” — because a universal law is the only one capable of commanding the
will through the mere thinking of its idea, independent of all sensible ends
and desires. (Hunter: 919-21.)
Conclusion
If this analysis is right, Kant does not 'begin [his] task anew' in Part II. Rather, if Part I is meant to persuade us from the side of ordinary moral thinking that we are all latent bearers of the moral will, through the idea of doing one's duty for its own sake, Part II reprises the idea that the categorical imperative involved in 'duty for its own sake' is also the principle of action of a purely rational will acting, unlike a possible God, under 'subjective limitations and hindrances'.
