Q:

What would it mean for morality not to be a consequence of rationality?

How would one manage to arrive at this morality, or when to apply it without reason?
Some might say we directly experience morality, but this is inconsistent with the different different moral intuitions throughout history. At best, we then need to use reason to work out what in our experience or moral intuition is genuine and what is just a societal by-product. e.g. One might say an Aztec could work out murder is wrong but then need to apply reason to deduce that human sacrifice is also wrong, even though many Aztecs thought it was a good action.
This is a follow-up question to my previous question is immorality just irrationality?

A:

How would one manage to arrive at this morality, or when to apply it
  without reason?

This really depends on what kind of metaethical view it falls under. Firstly, let's look at which positions are possible for externalism. (Externalism about metaethics is the view that knowledge of moral truths doesn't itself motivate.)
Smith presents it as a trilemma:
(1) Morality is objective.
(2) Morality is motivating.
(3) We are motivated by our desires. (Humean theory of motivation)
We can't believe all three. Typically expressivism denies 1, naturalism denies 2, non-naturalism denies 3. I believe other anti-realist views could deny multiple things.
So what seem to be looking for is: How would a view that denies 2 but affirms 1 look and how could we find out what's moral under it? There are multiple possibilities. Here's one:
This is Brink's view in Moral Realism and the Foundations of Ethics. He defends externalism and (non-reductive) naturalist moral realism, so he fits in what you're looking for. How does he think we can arrive at moral truths? Brink goes for a coherentist epistemology, so if our beliefs are coherent then they are justified. If we want to find out moral truths then we should try to make all our beliefs coherent. He has a whole chapter on it, so you could check it out.
How would that look in application? One concept to do so would f.e. be the reflective equilibrium (I don't think Brink goes into it, but it goes for coherentism about moral beliefs in general).

Some might say we directly experience morality, but this is
  inconsistent with the different different moral intuitions throughout
  history.

Not necessarily or at least there are possible defending arguments. Moral intutionists could argue that people were mistaken and contradicted their own intuitions, or that many immoral beliefs can be explained by wrong factual beliefs.
If you're looking for "How do you arrive at morality?" in general (and not just for externalist views) then check out this article and paragraph 5 of this article.

