The phrase “the goose hangs high” means things are propitious; all is well.
This is sometimes regarded as a corruption of “the goose honks high,” on the supposition that, in fair weather, the geese fly high and honk as they fly.
The editors of the Dictionary of American English, however, report that they find “no convincing evidence” of such a corruption. Dr. Frank H. Vizetelly, in his Deskbook of Idioms and Idiomatic Phrases (1923), considers that the expression alludes to the one-time cruel American sport of gander pulling.
As described in 1818 by Henry B. Fearon, in his Sketches of America, “This diversion consists in tying a live gander to a tree or pole, greasing its neck, riding past it at full gallop, and he who succeeds in pulling off the head of the victim, receives the laurel crown.”
Presumably Dr. Vizetelly thought that if the gander (or goose) was tied to the tree high enough, the contestants were assured of good sport.