The Australian-English phrase more hide than Jessie and its variants denote an excess of effrontery.
—Cf. also early Australian uses of ‘more front than’ and British and Irish uses of ‘more front than’.

The phrase more hide than Jessie:
– puns on two meanings of the noun hide: the literal meaning, i.e., the skin of an animal, and a figurative meaning, i.e., effrontery;
– refers to Jessie, the name of an elephant that was kept first at Moore Park Zoological Gardens, then at the Taronga Park Zoo, both in Sydney, New South Wales.

On Saturday 1^st December 1883, The Sydney Daily Telegraph (Sydney, New South Wales) announced that Jessie had arrived at Sydney the previous day:

Yesterday morning the steamer Newcomen, from Calcutta via Melbourne, arrived at the Central wharf, Miller’s Point, having on board the female elephant and a number of mongeese, all of which are at present at the Zoological Gardens, Moore Park. […]
The female elephant will be known as Jessie, and her age is given as sweet seventeen, and her capacity for negotiating sugar cane and other eatables is put down at £6 per week. She is a remarkable contrast to Jumbo, being a good deal smaller and exceedingly tractable, and as such will be a glorious treat for the children.

Jessie died on Tuesday 26^th September 1939. The following photograph and caption are from The Sydney Morning Herald (Sydney, New South Wales) of Wednesday 27^th September 1939:

Jessie, the 77-year-old elephant at the Taronga Park Zoo, who died yesterday, photographed at her last birthday party. More than 1,500,000 children had ridden on her back.

These are the earliest occurrences of the phrase more hide than Jessie and of its variants that I have found, in chronological order:

1-: From The Newcastle Sun (Newcastle, New South Wales) of Saturday 4^th January 1919:

STOCKTON FERRIES

“Xenophon” writes:
Permit me to direct attention of the weekly ticket holders of the Newcastle-Stockton ferries to the notice in the waiting shed, about the increase in price of tickets. The antiquated carts, which they have the hide of “Jessy” to call ferry boats, are a disgrace to the Newcastle district. In wet weather it is the usual thing to find the water dripping down on the seats, especially the seats of the smoking compartment, which are generally occupied by the gentler sex. It is nearly time something was done to rectify the existing state of these ferries. Perhaps some of the aldermen will take up this matter.

2-: From the Labor Daily (Sydney, New South Wales) of Tuesday 10^th November 1925:

“MORE HIDE THAN JESSIE”
NAT. ORGANISERS
HOW THEY PASTE BILLS

Nationalist candidates and organisers, are everywhere noted for having more hide than Jessie, the Zoo elephant.
But the desperate straits they are in at the present time have resulted in stunts by them which would cause Jessie to lose “that schoolgirl complexion” in sheer envy.

3-: From The Cumberland Argus and Fruitgrowers’ Advocate (Parramatta, New South Wales) of Friday 19^th August 1927:

“HIDE”
Health Department Criticised
Creek Flushing Job

“The Health Department has more hide than ‘Jessy’ to ask us to flush out this creek. It’s all right for Dr. Purdy to sit in his office and say we have to undertake such a job, just as though it were a gutter,” stated Alderman Dunbar at the Lidcombe Council on Wednesday night, when a report was received from the Health Department relative to the condition of the drain in Swete-street, which the Department claimed should be flushed out.

4-: From a statement made by one Robert B. S. Hammond, published in the Newcastle Morning Herald and Miners’ Advocate (Newcastle, New South Wales) of Friday 24^th August 1928:

a personal chat
with the Returned Soldiers

Dear Diggers,—
Dinki-di. Old Man Booze has more hide than Jessie at the Zoo. It makes my blood boil to see one particular poster on every publican’s windows.
It displays the words—“WE FOUGHT FOR LIBERTY . . . HOLD IT”—with a picture of a Digger in the centre of the poster.
Just what do these birds mean? Do they mean that THEY, the PUBLICANS, fought for Liberty?

5-: From The Uralla Times (Uralla, New South Wales) of Monday 9^th September 1929:

Elephant Jessie Misbehaved
SWALLOWED COAT OF A LADY.

Fifty-six years old and she ought to know better.
The proverbial ‘hide of Jessie’ was demonstrated the other day at the Sydney Zoo by Jesssie [sic] herself .
A woman carrying her coat over her arm stood in the shadow of the elephane [sic] garage while Jessie was being loaded up with kids.
Whether the lady said anything cattish or nasty to the elephant is not known, but Jessie’s trunk reached out and the coat is now inside Jessie.

The variant more cheek than Jessie occurs in the following passage from Jimmy Brockett: Portrait of a notable Australian (London: Britannicus Liber, 1951), a novel by the Australian author Dallas George ‘Dal’ Stivens (1911-1997):

“Find me a good furnished flat near the Cross,” I told young Herb later. “And if you blab about it to anyone I’ll screw your bloody neck!”
Herb winked at me. […]
“What’s up, boss? Fallen out with the trouble and strife?”
“You’ve got more cheek than Jessie the elephant!” I said. “Do as you’re told and don’t ask any questions.”

And the variant more arse than Jessie occurs in this passage from The Long White Night (London: Frederick Muller, 1965), a novel the Australian author by Eric Lambert (1918-1966)—as quoted by Gerald Alfred Wilkes (1927-2020) in A Dictionary of Australian Colloquialisms (Sydney University Press in association with Oxford University Press Australia, 1990):

‘You’ve got more arse than Jessie,’ I told him.