String cites: how to order signals and cases (and cite two cases in one sentence)

Short answer. Group the authorities by signal, supporting signals first. Supporting signals share one citation sentence, with semicolons between authorities. A contrary signal such as but see starts a new citation sentence. Within one signal, lead with the strongest authority. If two parts of your sentence rest on different sources, cite each one right after the part it supports, as a clause set off by commas.

The rules behind it
[Convention] Signals come in four groups: support (no signal at all, plus e.g., accord, see, see also and cf.), comparison (compare ... with ...), contradiction (contra, but see, but cf.) and background (see generally). They appear in that order.
[Convention] In a citation sentence, signals from one group sit together, separated by semicolons; a signal from another group opens a new citation sentence. A citation clause is different: it can hold every group, still separated by semicolons.
[Convention] A signal covers every authority after it until the next signal, so you do not repeat see before each case.
[House style] Inside one signal, start with the authority that carries the most weight for your point. After that we go enacted law, then cases, then secondary sources; binding cases before persuasive ones; and newer before older within one court. If the court you are writing to prescribes an order, use its order. A free public-domain citation manual asks only for a "helpful and logical" order.
[Convention] Keep strings short. Add an authority only when it gives the reader something the others do not, such as showing a split.

A real string, read slot by slot
On page 413 of Loper Bright, the 2024 decision that overruled Chevron (cited in full below), Justice Thomas's concurrence opens a citation sentence with See and two of his own earlier opinions, separated by a semicolon, each with its own label for the kind of opinion. A semicolon and see also then add a third. All three signals support his point, so they share one citation sentence. A but see would have started a new one.

One signal over several authorities
Earlier on the same page, the concurrence cites 5 U.S.C. § 706 with no signal, because the statute itself says what he describes, then writes see also once and lists two authorities after it: a range of pages in the majority opinion (ante) and a dissent from the denial of certiorari. The one signal covers both, and the enacted law comes first.

Two cases in one sentence
When each half of a sentence has its own support, a string at the end blurs which case supports which half. Use citation clauses instead:
An agency told to regulate when doing so is "appropriate and necessary" has to pay some attention to cost, Michigan v. EPA, 576 U.S. 743, 752 (2015), and a court may not defer to the agency's reading of a phrase like that just because the statute is ambiguous, Loper Bright Enters. v. Raimondo, 603 U.S. 369, 412–13 (2024).
The Supreme Court does the same on page 405 of Loper Bright. One sentence says it has not applied Chevron to judicial-review provisions, with a see clause citing one case, "or to statutory schemes not administered by the agency seeking deference," with a see clause citing another.
[House style] Otherwise, default to one citation sentence after the sentence it supports.

Common mistakes
But see after a semicolon in the middle of a supporting string.
See repeated before every case under one signal.
A ten-case string where two strong cases would do.
Ordering by date or alphabet when one case is plainly the strongest.
A string at the end of a sentence whose halves rest on different authorities.

Read more
What each signal means, with a real example of each: https://citationcode.com/examples/citation-signals (opens in a new tab)
The Anatomy chapter (its opening is free to read): https://citationcode.com/book/anatomy (opens in a new tab)