Only partially known.
The only sense in which this tourney was 'continued'
in New York Clipper, as stated by White
p. 17, seems to be publication of winning problem. In the absence
of a tournament report or access to New York Saturday Courier for the relevant period, it is difficult to be certain if that is correct.
On 1856-10-11 the problem below was published in Clipper as problem 25 with the author’s name and the title “A prize problem”. In the Loyd-book, White adds an epigram that may be a tourney motto ("A Bold Attempt is Half Success"), but the source for it is not known.
The following week, the editor (Hazeltine) added: “This gem of chess strategy was received by us, but not published, while connected with another paper. It was in competition for a prize, and for this problem our ingenious young contributor will receive that prize the first Wednesday evening on which he can make it convenient to attend our private chess soirees.” (No direct references to New York Saturday Courier have been found in Clipper, but Whyld indicates that Hazeltine was previously editor for the New York Saturday Courier chess column. See Whyld: Chess Columns.) The New York Saturday Courier ceased publication on or after 1856-09-13 according to Whyld; US Library of Congress does not identify its end as definitely.