What is the origin of "smell a rat"?

The phrase "smell a rat" in the sense of "detect or suspect that something underhanded is going on" appears in Nathan Bailey, A Universal Etymological English Dictionary (1731):

To smell a RAT {soupconner, F. subolere, L.} to discover some intrigue.

John Ray, A Compleat Collection of English Proverbs, third edition (1737) lists "I smell a rat" as a proverbial phrase in English, although the first edition (1678) does not; I don't have access to the second edition.

The phrase also appears in Samuel Johnson's first Dictionary of the English Language (1756) and later editions, with a definition that is even closer to the modern sense of the term:

To smell a RAT. To be put on the watch by suspicion. Hudibras.

Johnson's citation of Samuel Butler's Hudibras (complete by 1684) refers to this passage:

Quoth Ralpho, who could hold no longer,/Altho' I am no good States-Monger,/Yet I begin to smell a Rat,/And what your Worship would be at./You have, I find, some little Guilt,/For Christian Blood devoutly spilt;/Some inward Checks and Throws of Conscience,/Which strictly speaking, are all Nonsense,/And those have press'd you on so far,/That you resolve to quit the War.

This excerpt appears in Book Two of Hudibras, which was originally published in 1664, so we have an instance from that date of "smell a Rat" in the sense that Johnson identifies in his dictionary (and that remains current today).

Earlier still is this instance from "The British Bellman" (1648), reprinted in The Harleian Miscellany, volume 7 (1746):

We are in a pitiful Case now ; to stay or go we know not ; stay, and the Scots and the Lord Inchiquin come in upon us ; go, and the City follows us. I smell a Rat ; the blazing Comets are going out with a filthy Stink ; an Ordinance of Parliament to pass four great Ships without Search, laden with Money, and now at Gravesend, or newly put to sea.

But earliest of all in Google Books search results is this mention in Ben Jonson, The Case Is Altered (1597/1609):

Juniper ["a cobler"]. Do you hear, sweet soul, sweet radamant, sweet mathavel? one word, Melpomene, are you at leisure?

Rachel de Prie. At leisure! What to do?

Juniper. To do what! to do nothing, but to be liable to the extasy of true love's exigent, or so; you smell my meaning.

Onion [a servant]. Smell! filthy, fellow Juniper, filthy. Smell! O most odious!

Juniper. How filthy?

Onion. Filthy by this finger. Smell! smell a rat, smell a pudding. Away, these tricks are for trulls; a plain wench loves plain dealing; I'll upon her myself, smell to march-pain wench.

So "smell a rat" may be in use in its modern figurative sense as early as 1597.

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Earlier instances under the spelling 'smell a ratte [or ratt]'

Belatedly I checked for instances of "smell a ratte," and found two instances. From Anthony Gilby, A Pleasavnt Dialogve Betweene a Souldior of Barwicke, and an English Chaplaine (1566), quoted in The England and Holland of the Pilgrims (1905), a character named Sir Bernarde Blynkarde speaking:

"Me thinke I smell a ratte in this geare. All is not gold that glittereth."

And from 1560 (probably), though cited in a collection published in 1580, is Thomas Churchyarde, Farewell from the Courte, the Second Yere of the Queenes Majesties Raigne (Queen Elizabeth succeeded Queen Mary in 1558):

Me thinke you chuse your shoppe not well In court your follies for to sell: That shopp stands full within the winde, Or els so muche in peoples minde That if one fault be in your ware Tenne thousand eyes thereon doe stare; And when thei finde a counterfeite, Or see fine merchaunts use deseite, Thei crie a loude, Wee smell a ratte. Some have more witte within their hatte Then in their hedde; that sells such stuffe.

Both of these instances of "smell a ratte" are obviously figurative, and push the documented first occurrence in Google Books back to 1560–1566.

J.S. Farmer & W.E. Henley, Slang and Its Analogues Past and Present (1902), cite an instance of "smell a ratt" in a poem originally attributed to John Skelton and dated to 1529 (although other scholars put the date at 1540 and say that Skelton could not have been the author), titled "The Image of Ipocrysy":

For pleynly to be breve, So nye they [the clergy] do vs dreve, That we, to our great greve, Must say that white is blacke, Or elles they sey we smacke, And smelle we wote not what: But then beware the catt; For yf they smell a ratt, They grisely chide and chatt, And, Haue him by the jack, A fagott for his backe, Or, Take him to the racke, And drown hyme in a sacke, Or burne hyme on a stake!

Here the use of "smell a ratt" is immediately preceded by the appearance of "the catt," which suggests a literal sense of smelling the rat—and yet in the context of the poem, the cat represents the clergy and the rat represents folk of simple (unspecified) faith who may nonetheless run afoul of official church doctrine or dogma. So it isn't much of a stretch to read "smell a ratt" here as meaning something like "sense something heretical or untoward."