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The expression last-chance saloon designates:
– a final opportunity or hope for success;
– the last refuge of the unsuccessful or desperate.

This expression occurred, for example, in Conferences: How to Use a Checkbook, published in Time (New York City, New York, USA) of Monday 7^th July 1947 [page 23, column 2]—the following is about the Marshall Plan (i.e. the European Recovery Program, a plan for aiding the European nations in economic recovery after World War II, proposed by the U.S. Secretary of State George C. Marshall in 1947):

Last week in Paris, the plan found an enemy—Russia. […]
[…]
There remained only a faint possibility of compromise with the Russians. Paris had been the last chance for One Europe (a U.S. reporter had dubbed the Parrot Salon, where the Ministers met, the “Last Chance Saloon”). For those nations which wanted to take part, the U.S. could push the Marshall Plan ahead without Russia.

Of U.S. origin, the expression last-chance saloon originally designated a saloon at the edge of a town or at the border between two U.S. states. This expression often occurred in conjunction with first-chance saloon.

The following two texts illustrate this original meaning of the expression last-chance saloon:

1-: From Letter from Texas, dated Denison, Texas, Sunday 15^th December 1872, an account of a journey from Paola to Denison, by a person signing themself ‘Texas’, published in the Miami Republican (Paola, Kansas, USA) of Saturday 21^st December 1872 [page 2, column 3]:

We were ferried to Texas soil. […] As we left the northern shore, the “Last Chance” saloon faded away, but on the Texas shore the “First Chance” stood ready to receive us. Both “chances” drive a thriving trade, as many weary, worn wayfarers have emptied their flasks before they reach the end of the track. Some take one “chance,” some the other, and persons of arid constitutions take both. Some drink because of thirst; others because of the novelty of the thing, among the latter of whom is the undersigned.

2-: From Sam Jones, Gospeler, published in the Sunday Globe-Republic (Springfield, Ohio, USA) of Sunday 7^th June 1885 [page 6, column 5]—reprinted from the Chicago Herald (Chicago, Illinois, USA):

On a road that leads into Chattanooga there was once a saloon with a sign in front of it. It was just on the outside of the town. The sign read thus: “The First Chance.” And on coming out, the letters on the other side of the board were: “The Last Chance.”

These are, in chronological order, the earliest occurrences of last-chance saloon that I have found—this expression seems to have originated in California:

1-: From The Nevada Democrat (Nevada City, California, USA) of Wednesday 27^th January 1858 [page 3, column 5]:

DELINQUENT TAX LIST.
NEVADA CITY.

TO WHOM ASSESSED. DESCRIPTION OF PROPERTY. TAX DUE.
Ashmore D W—Arcade saloon and lot Broad st, and 2 small houses in rear $11 00
[…]
Jeffreys, James—Bar fixtures in Last Chance saloon also dwelling house and lot on old Wash’ton road, 10 00

2-: From The Hydraulic Press (North San Juan, California, USA) of Saturday 14^th April 1860 [page 2, column 5]:

Sam. Abbey and Geo. Cullodi excite vivacity by their invitations to bibacity in an advertisement under the head of “First and Last Chance Saloon.”

3-: From The Hydraulic Press (North San Juan, California, USA) of Saturday 21^st April 1860 [page 1, column 1]:

SALOONS & RESTAURANTS
The First and Last Chance Saloon

THE PUBLIC IS INFORMED THAT ABBEY & CULLODI have opened a new and handsome

LIQUOR AND SMOKING SALOON

in the building formerly known as the Pioneer Liquor Store. They will keep the purest Wines and Liquors, best Cigars, and most fascinating Ba. keepers.
Everybody who took the first chance in California, the second at Frazer river, or the last in

Washoe,

are invited to give the subscribers a call.

North San Juan, April 14, ’60.
SAM. ABBEY,
GEO. CULLODI.

4-: From Changes, published in The Hydraulic Press (North San Juan, California, USA) of Saturday 10^th November 1860 [page 3, column 1]:

The “First and Last Chance Saloon,” of Abbey and Cullodi, has been removed one door above the former stand.

5-: From The Hydraulic Press (North San Juan, California, USA) of Saturday 16^th February 1861 [page 3, column 1]:

Plank the Streets.—If the citilens [sic] owning property on the south side of Main street, west of the “Last Chance Saloon,” would plank the street in front of their buildings, it would make a great improvement in that portion of our village. Plank the streets, say we.

6-: From State News, published in the Buchanan County Bulletin (Independence, Iowa, USA) of Friday 4^th June 1869 [page 2, column 2]:

Harvey Patney of Des Moines got drunk, was run over by his wagon, and died from his injuries. Just before his death he sent the following message to the editor of the Bulletin:
“I want you to say in your paper that the liquor I became intoxicated on, was obtained by me in South Des Moines at the Last Chance Saloon; that I paid for seven glasses, drank three, and threw away four of them. I do not believe but that it was poisoned, because it affected me so soon.”
Was not the keeper of that saloon a murderer?

7-: From Local Intelligence, published in the Daily Press and Herald (Knoxville, Tennessee, USA) of Friday 11^th June 1869 [page 4, column 1]:

Last Chance Saloon.
The Last Chance Saloon has been refitted and well stocked with liquors.
M. Kennedy, Agent.

The following early allusive use of last-chance saloon is from Proceedings of the Annual Session of the Nebraska State Pharmaceutical Association (Lincoln (Nebraska): State Journal Company, Printers) of 1888 [Volume 7, page 28]:

The eye of the public is upon him; if he hesitates he is incompetent, though he may only be seeking a way out without exposing the writer of the prescription, who has made a blunder. You have heard of last chance saloons, but every druggist—not every drug store—is the last chance in the sense that he often stands between his patrons and a fatal mistake.