Typically, or at the very least, to my basic understanding, Ethic's theories give a definition of what constitutes as Ethical, but does not say how to discern which decisions are more ethical relative to another. For instance, if we are to consider the golden rule (or, more fancily, Schopenhaur's reduction of Kant's ethics), then we'd find that it only tells us what would be ethical.
Are there any theories on how to turn the set of possible ethical options into a partially ordered set?
The fundamental problem with your question is that there are multiple dimensions associated with real life ethical decisions. To create an ordered set, you would effectively be trying to model a function of multiple variables by a function of one variable. It would be like trying to create an ordered set of foods based on taste. There are too many different components of taste to summarise with a single value that can be used to determine a unique order.
Consider a classic trolley problem. A trolley is about to kill two people. If you act you can save either one of them. How would you decide? Suppose one was young, and one old- would that matter? Suppose one was blind- would that matter? Suppose one was a known criminal- would that matter? Suppose one was male the other female- would that matter? Suppose one was in a wheelchair- would that matter? Suppose one had an incurable decease- would that matter? Suppose one believed in an afterlife and the other didn't- would that matter? suppose one had five dependents and the other had three- would that matter? Suppose one worked as a nurse- would that matter?
It should be obvious that with more time and very little thought I could have extended that list to any number of other independent variables that you might have to take into account to decide which of the two people you ought to save. And now consider that in deciding between the two people, you might have to take combinations of the variables into account, since several of them could apply at the same time. It is simply not possible to reduce the complexity of the choice to an ordered list.
In §7 of A Theory of Justice (1999 ed.), John Rawls writes:
I shall think of intuitionism in a more general way than is customary, namely, as the doctrine that there is an irreducible family of first principles which have to be weighed against one another by asking ourselves which balance, in our considered judgment, is the most just ... [and as the view that] there is no single standard that accounts for them or assigns them their weights. ... if there are priority rules [emphasis added], these are thought to be more or less trivial and of no substantial assistance in reaching a judgment.
The next section is entitled "THE PRIORITY PROBLEM" and here Rawls brings in what he calls a lexical order, or what is usually called in mathematics proper a lexicographical order: "This is an order which requires us to satisfy the first principle in the ordering before we can move on to the second, the second before we consider the third, and so on." He speaks of "a sequence of constrained maximum principles," which is a turn-of-phrase reminiscent of Zorn's lemma, which is known to be usually interchangeable with the axiom of choice that usually grounds the well-ordering principle in transfinite set theory. Rawls' sensitivity to set-theoretic and related concepts/descriptions will be showcased throughout, here, and so it is doubtful that he was not at least subconsciously aware of that similarity in phrasing.
Now, Rawls' confidence in the lexical order thesis is not absolute: "... it seems clear that, in general, a lexical order cannot be strictly correct" (§8). He will go on to mention significant difficulties in squaring animal-welfare/environmentalist ethics with the humanistic theory of justice (§77), and even when human equilibrium is all that is at stake, he warns of a possible "day of reckoning" for unstable societies (§46), especially in light of the "severe if not impossible tests" that questions of justice-between-generations and time preference pose for ethical theories (§44/45). Therefore, although he is confident that his theory of justice is worth evaluating, he is not overly committed to its completeness, internally or externally, and the incompleteness in play can undermine the viability of the ordering he proposes.
That there should be some such ordering available, though, he also stipulates as part of "the formal constraints of the concept of rightness" (§23):
... a conception of right must impose an ordering on conflicting claims. ... There is a difficulty, however, in deciding what counts as an ordering. It is clearly desirable that a conception of justice be complete, that is, able to order all the claims that can arise (or are likely to in practice). And the ordering should in general be transitive... [moreover, there is a requirement of] finality. The parties are to assess the system of principles as the final court of appeal in practical reasoning [which covers] the totality of relevant considerations and their appropriate weights...
He argues that the cardinal-value judgments expressive of utilitarianism can be dispensed with by instead using the image of "the least advantaged representative citizen" as the minimal element (so to speak) in the economic ordering (§15):
... as long as we can identify the least advantaged representative man, only ordinal judgments of wellbeing are required from then on. We know from what position the social system is to be judged. It does not matter how much worse off this representative individual is than the others. The further difficulties of cardinal measurement do not arise since no other interpersonal comparisons are necessary.
(§18) includes a tree whose branches are the "location" of various moral terms in a set of sequences from "practical reasoning" at the root of the tree and the priority rules as the sunlight in which the tree grows. He generally has rules for groups of people as prior to rules for individuals; the tension between the individualistic aspects of his theory, and the need for a socialized ethos, he goes back and forth on leading up to (§79). But so even for individuals, he claims (§51) that our point-of-departure responsibility is to uphold the background system of justice when this is available and to the extent that supporting it is feasible. However:
I do not know how this problem [of ordering individual duties] is to be settled... I shall not attempt to discuss these questions of priority in full generality. What I shall do is to examine a few special cases in connection with civil disobedience and conscientious refusal [objection to military induction] under circumstances of what I shall call a nearly just regime.
I should note that he explicitly avoids trying to enumerate all possible ethical questions and theories (§21), arriving at the priority of liberty over the difference principle based on comparing various historically recognized pairs and tripletons of claims.
Now, Rawls is not the only theorist whose arguments tend towards a relatively precise weighting of options. Immanuel Kant, for example, distinguishes so-called perfect from imperfect duties and seems to prioritize the former over the latter, at least in abstracto; this distinction is a subsidiary of the narrow/wide distinction, which he brings up in the Doctrine of Virtue (The Metaphysics of Morals, 211[411]) in this connection:
But, it will be asked, why do I introduce a division of ethics into a Doctrine of Elements and a Doctrine of Method, when no such division was needed in the doctrine of Right? The reason is that the doctrine of Right has to do only with narrow duties, whereas ethics has to do with wide duties. Hence the doctrine of Right, which by its nature must determine duties strictly (precisely), has no more need of general directions (a method) as to how to proceed in judging than does pure mathematics; instead, it certifies its method by what it does. But ethics, because of the latitude it allows in its imperfect duties, inevitably leads to questions that call upon judgment to decide how a maxim is to be applied in particular cases, and indeed in such a way that judgment provides another (subordinate) maxim (and one can always ask for yet another principle for applying this maxim to cases that may arise). So ethics falls into a casuistry, which has no place in the doctrine of Right.
We might also think over Thomas Aquinas' cardinal virtues, "from which Aquinas’s more extensive list flows" towards the final end of justification and sanctification, the visio beatifico of God.
Harry Gensler's so-called "formal ethics" is yet another endeavor of the kind you might be asking after. Informed by the 20th-Century theme of finely-parsed axiom-to-theorem lists, this system might be able to deliver weightings on a case-by-case basis, depending on how precise its premises and inference rules are (or can be made to be). And so on and on.
Granted, there's no universally compelling, or even fully self-understood, theory of ethics. Particularists and other theorists have rejected attempts at overbearing systemization, no less (see e.g. Bernard Williams' objections to what he calls "the peculiar institution").
And so there is arguably one of the subtlest in-jokes in philosophical history, here, when Rawls says (§9), "Intuitionism is not constructive," seeing as mathematical intuitionism is typically paired with mathematical constructivism.
