A long time ago I was looking at intrinsic topologies for ordered structures and I came across a paper by Jimmie Lawson that catalogued all kinds of topologies that could be defined intrinsically on (semi)lattices. Here are links to reviews on zbMath of that paper and a related one. The latter shows that the topology of a compact topological (semi)lattice is intrinsic and unique. You'll see there's many ways to topologise a (semi)lattice; some of the ways work on partial/quasi-orders as well.

You can also find it as Lemma 4.5 in Jech's The Axiom of Choice.

In the original paper, Über die Unabhängigkeit des Wohlordnungssatzes vom Ordnungsprinzip, Fund. Math. 32 (1939), 201-252 you'll find this in statement 101 on page 243. But there is a lot of preparatory work to go through before that.

@MikhailKatz It seems that for many analysts the Axiom of Choice was unproblematic, see for example Banach's proof of the Hahn-Banach theorem in Studia Mathematica, 1 (1929), 211-216. The one-step extension is proven as Theorem 1, the full theorem is Theorem 2: On prouve ce théorème par induction transfinie en appliquant succesivement le théorème 1 aux éléments de l'ensemble $E-G$ (supposé bien ordonné).

@MikhailKatz I looked but could not find anything. Rudin certainly knew his Set Theory, but I think he was completely comfortable with the Axiom of Choice. In the proof of the Hahn-Banach theorem in Functional Analysis he does the one-step extension and then says that "The second part of the proof can be done by whatever one's favorite method of transfinite induction is; well-ordering, Zorn's lemma, or Hausdorff's maximality theorem:" He uses the last one. In the appendix he indicates where that theorem is also used in the book: Krein-Milman and the maximal-ideal theorem in rings.

Jones' paper is quite informative too on what is possible with discontinuous additive functions and their graphs.