This is a familiar issue to me, but I also have not found it anywhere in the literature or standard references, although I have usually dealt with the conditional, rather than an accessibility relation.
It does not seem to be widely recognized that the material conditional of classical two-valued logic is just such a partial ordering, (That is, a relation which is reflexive, transitive, and anti-symmetric) and furthermore, that it is chiefly what makes deductive reasoning with extended chains of claims possible.
This is also a difficulty in multi-valued logic. An examination and survey of the many-valued systems in the standard references on many-valued logic reveals that most of the conditionals defined are not partial orderings and do not have the desired properties.
I am most familiar with the conditional relation of Lukasiewiz 3-valued logic which also fails to have these properties, Hovever, within this system it is possible to define a "definite" or "strict" conditional that does. I have not found any references which employ this approach.
The Lewis systems and their kin typically employ the strict conditional symbolized by "fishhook" as the connective of choice, but it is not clear at first glance whether it has the desired properties.
However, in the 3-valued system, it is also possible to define a conditional which satisfies the usual definition of the Lewis strict conditional and which fails as a partial ordering.
It may be interesting to note that if the Lewis strict conditional holds, so does the Lukasiewicz strict conditional, but not conversely. This raises the possibility that a truth-functional system of modal logic can be constructed on the basis of 3-valued logic.
This possibility is generally dismissed in the literature I have examined as having been found unworkable, but the reasons, if they are given at all, don't hold up to close examination.