Coined during the interwar period, the Liverpudlian ¹ dockers’ phrase three on the hook (and) three on the book and its variants mean three days at work (and) three days on the dole ².

In this phrase, hook refers to the dockers’ tool, book refers to the unemployment register.

¹ The adjective Liverpudlian means of, or relating to, Liverpool, a city and seaport in north-western England, situated at the east side of the mouth of the River Mersey.
² Since the end of the First World War, dole, usually preceded by the definite article the, has been the British-English popular name for the various kinds of weekly payments made from national and local funds to the unemployed.—Cf. the British use of ‘dole’ and origin of the Scottish and Irish phrase ‘on the buroo’ (‘on the dole’).

In The Autobiography of a Liverpool Irish Slummy (New York: Vanguard Press, 1933), Pat O’Mara (1901-1983) explained the system denoted by the phrase three on the hook (and) three on the book:

With the dole abject starvation was out of the question. One could work three days at the dock for thirty-six shillings and draw the three days dole at three shillings per day to boot, making a grand total of forty-five shillings for the week of three days actual work! If one worked four days at the dock, there would be no dole, so the trick was to see to it that only three days were worked. And if no work were forthcoming, there would be eighteen shillings anyway. This beneficence was amazing when my mother compared it with former years.

These are, in chronological order, the earliest occurrences of the phrase three on the hook (and) three on the book that I have found:

1-: From a letter to the Editor, published in the Evening Express (Liverpool, Lancashire, England) of Tuesday 28^th May 1940:

Every Day a Pay Day
In regard to the signing on and off system at the docks, there is only one sensible method, which should have been adopted years ago.
Every docker should be paid unemployment pay every day he is idle, and the clearing houses could do much to alleviate the present “three on the hook and three on the book method.”—J. J. B., Liverpool.

2-: From the Liverpool Echo (Liverpool, Lancashire, England) of Friday 21^st August 1942:

Mr. Fred Cripps, dock manager, Elder Dempster Lines, in a letter […] to the Times, says:—
[…]
After three years of total war company directors are still thinking in terms of profit. The dockers still consider themselves casual labourers, and the mentality of some of them is still three days on the book and three days on the hook. The trades union officials still cling to the so-called privileges won in the days of peace, privileges which, in fact, are largely responsible for the Port of Liverpool being so badly equipped.

3-: From Strange Charm Of The Lingo Of Liverpool’s Dockland, by Frank Shaw, published in the Liverpool Echo and Liverpool Express (Liverpool, Lancashire, England) of Thursday 30^th July 1959:

The work is not easy and goes on in all weathers. As with the office worker’s elevenses short breaks are essential. But if a docker goes too far in this direction he will find himself stood off altogether—such spells off are “werkin for the Queen” and used to be called “three on the hook” the dockers’ tool of trade and “three on the book” that is signing on at the Labour Exchange. Sticter [sic] discipline may mean a longer spell off during which the Port Labour official “sits on the offenders book” pronounced “bewek” of course.

4-: From The Hungry Years, by Philip Key, published in the Liverpool Echo (Liverpool, Lancashire, England) of Monday 18^th January 1971:

They will always be remembered as the “Hungry Thirties” by thousands of people. A decade when jobs were scarce, unemployment was counted in millions, and the workless took to the streets.
[…]
Those bad days on the docks are remembered by many on Merseyside; men like a 70-years-old ex-Liverpool docker now retired. He is enjoying his retirement, and just wanted to be known as Bill.
Bill was among the thousands out of work during the Twenties, but during the tail end of that decade, he did secure a job on the docks.
He got his union card as a fruit porter—a job that did not need a “tally”—but later went on to become a stevedore and get that all-important “tally.”
The system in those days was to work half-days. “You could work for a week for 11 different employers,” recalls Bill. “I generally managed to get in two days’ work in a week, which was about average.”
He adds: “There was a system we called ‘three on the hook, and three on the book’—that was three days working and three on the dole.”
In those days the men stood around for the jobs and a tap on the shoulder meant you had one—for half a day. And then it was looking for work again.