"Mortal coil" explained

I heard the phrase this past week, implicitly citing Hamlet's soliloquy -

To be, or not to be, that is the question:
Whether 'tis Nobler in the mind to suffer
The Slings and Arrows of outrageous Fortune,
Or to take Arms against a Sea of troubles,
And by opposing end them: to die, to sleep
No more; and by a sleep, to say we end
The heart-ache, and the thousand Natural shocks
That Flesh is heir to? 'Tis a consummation
Devoutly to be wished. To die to sleep,
To sleep, perchance to Dream; Ay, there's the rub,
For in that sleep of death, what dreams may come,
When we have shuffled off this mortal coil,
Must give us pause...

- but after having heard it a hundred times, I wondered about the origin.  Here's the summary from Wikipedia, citing the OED:

Derived from 16th Century English, "coil" refers to tumults or troubles. Used idiomatically, the phrase means "the bustle and turmoil of this mortal life." "Coil" has an unusual etymological history. It was coined repeatedly; at one time people used it as a verb to mean "to cull," "to thrash," "to lay in rings or spirals," "to turn," "to mound hay" and "to stir." As a noun it has meant "a selection," "a spiral," "the breech of a gun," "a mound of hay", "a pen for hens", and "noisy disturbance, fuss, ado." It is in this last sense, which became popular in the 16th century, that Shakespeare used the word.