No, not published little by little. It was written in the jail in bits and pieces, assembled, and then published: first in part, then in full.
In King’s preface, when he says “the letter was continued on scraps of writing paper”, he means that the letter was written on various pieces of paper which were available to him. He is not characterizing the process of publication, just the process of writing.
The Wikipedia article “Letter from Birmingham Jail” states that the letter was written April 16, 1963 on the paper available to King. King “gave bits and pieces of the letter to his lawyers”, and these pieces were compiled and edited by the Reverend Wyatt Walker.
As for the publication of the letter, Wikipedia says:
Extensive excerpts from the letter were published, without King’s consent, on May 19, 1963 in the New York Post Sunday Magazine. The letter was first published as “Letter from Birmingham Jail” in the June, 1963 issue of Liberation, the June 12, 1963, edition of The Christian Century, and in the June 24, 1963, issue of The New Leader. It was reprinted shortly thereafter in The Atlantic Monthly. King included the full text in his 1964 book Why We Can’t Wait.
Incidentally, the full text of the letter is available online[PDF] courtesy of the Martin Luther King, Jr. Research and Education Institute at Stanford University.