Hello!
I absolutely adore the slow unfolding of "What the fuck"
In regards to the ending: at first, I thought that there was a body in the box, someone named June. This would be an interesting way to spin it, having all of this which feels supernatural culminate to something horrifically mundane.
I think what you are aiming for is more to do with time. One thing I noticed that was off then is "'That’s odd,' said the postman. 'Is it still July?'" If it's *nearly* June, shouldn't it still be May? Similarly, “That was sent to you in June," it was sent in *nearly* June.
I think that "The former postman" moniker should be consistent. It would make the line "Is a former postman still the postman?" stand out more.
The capitalization of "They" is interesting and I urge you to think about the "why" in choosing to do so.
I suggest doing a formatting overhaul to make the text look nicer and more put together. Consider breaking up paragraphs; a block of text all smushed together can be overwhelming for a reader.
Maybe you would like to indent the beginning of paragraphs:
[[div style="text-indent: 25px;"]]
Paragraph here
[[/div]]
A lot can be learned through re-reading your own piece; you notice the things that don't quite work and things that are inconsistent (such as the guest being brought tea but taking a sip from his coffee).