What does “oily-Gammon” mean?

Question

I have hard time understanding the term oily-Gammon in this sentence from How To Debate Politics Like A Gentleman.

Teddy Roosevelt would frequently lose his cool during debates on the Assembly floor. He’d call his opponents “cold blooded, narrow-minded, prejudiced, obstinate, timid, old psalm singing Indianapolis politicians” or “oily-Gammon, churchgoing specimens,” or simply “classical ignoramuses.”

The only meanings I can find for gammon are the following:

1. Ham which has been cured or smoked like bacon

2. A victory in backgammon (carrying a double score) in which the winner removes all their pieces before the loser has removed any

3. Nonsense, rubbish

What does oily-Gammon mean?

Answer 1

Oily gammon means literally fatty ham, and it's still in use in that sense:

Since I don't want a fatty or oily gammon, i grilled it first for a few minutes until it looks cooked :)) haha — Bubblews 12/29/2012

But Roosevelt is probably using it appellatively, referring to the arch-villain of a novel which was enormously popular in the nineteenth century, Ten Thousand A-Year, by Samuel Warren (1839). “Oily Gammon, Esq.”, partner in the law firm of Quirk, Gammon and Snap, became for two generations the archetype of smooth-spoken self-serving hypocrisy:

He lolled in his capacious arm-chair ; his hands meekly crossed ; every individual hair smooth and shining above the brow he thought so fine ; a white cravat bounding his fleshy chin ;—a large-sized picture of sanctimoniousness or hypocrisy ; an over-fed, church-going Oily Gammon. —Marion Harland, The Hidden Path, 1856.

Under a gruff exterior there is often a kindly heart, and who has not known an Oily Gammon who, in spite of his smooth ways and kindly speeches, has turned out to be utterly heartless and selfish? —Current Literature, 1893.

[John D. Rockefeller] became exceedingly social in his work, gradually making the acquaintance of all the widows and orphans in the middle West, and is now known as the Oily Gammon of the world at large. —Life, vol. 49, 1907.

Answer 2

Besides the meanings of gammon that you mention, OED1 (1901) shows a nautical meaning (“The lashing of the bowsprit”) and several meanings that came originally from thieves’ slang or colloquialisms. Among these is:

Gammon (gae-mon), sb.4 slang or colloq. Also 8 gamon. ...
3. Ridiculous nonsense suited to deceive simple persons only; ‘humbug’, ‘rubbish’. ... 1805 T. HARRAL Scenes of Life III. 105 ‘Come, come, none of your gammon!’ cried one, ‘tell us where the other black sheep is’.... 1845 DISRAELI Sybil (Rtldg.) 285 Morley has got round them, preaching moral force, and all that sort of gammon.

The senses of oily include “Excessively friendly or polite so as to sound insincere”. It may be that Roosevelt referred to the “Oily Gammon, Esq.” mentioned in StoneyB's answer, or he may have harkened to older meanings (such as “The ham or haunch of a swine” and “Ridiculous nonsense suited to deceive simple persons only”) on which the name “Oily Gammon, Esq.” was based.