The idea here is of thrusting your hands deep into your pockets to find money with which to pay for something.
1991 - Sports Illustrated - You really have to dig deep night after night to get up for every game.
RELATED IDIOMS :
go off the deep end = go in off the deep end
give way immediately to anger or emotion - informal
This expression refers to the deep end of a swimming pool where the diving board is located. In the USA the phrase has also developed the meaning go mad…but in either sense the underlying idea is of a sudden explosive loss of self-control.
in deep waters
in trouble or difficulty – informal
In deep water is a biblical metaphor. See for example, Psalm 69 :14 : let me be delivered from them that hate me and out of the deep waters.
jump in at the deep end = be thrown in at the deep end
face a difficult problem or undertaking with little experience of it - Informal