Green’s Dictionary of Slang

high jump n.

[? steeplechasing; or of trouble so bad that one will metaphorically have to ‘jump very high’ to get over it; but ? image of death by judicial hanging]

1. [late 19C–1940s] the gallows, thus death.

2. [20C+] serious problems.

3. [1950s+] (Aus.) the criminal court.

In phrases

in for the high jump (also in for the jumps, on the high jump)

[1920s+] in serious trouble.