Green’s Dictionary of Slang

handkerchief n.

In phrases

give someone the handkerchief (v.)

to give someone a signal.

1989

in G. Tremlett Little Legs 194: handkerchief, as in ‘gave him the handkerchief’ to give one a signal, deriving from the common practice among criminals of making prearranged signs to each other in crowded places by the use of a handkerchief.