(A), in George III.'s time, meant a flannel petticoat. It was afterwards applied to what were called false shirts —i.e. a shirt front worn over a dirty shirt, or in lieu of a shirt. These half-shirts were first called Tommies.
A hundred instances I soon could pick ye - Without a cap we view the fair, The bosom heaving also bare, The hips ashamed, forsooth, to wear a dicky.
Peter Pindar: Lord Auckland's Triumph.
So again: -
And sister Peg, and sister Joan, With scarce a flannel dicky on ...
Middlesex Election, letter iv.
(Hair, whalebone, or metal vestments, called dress-improvers, are hung on women's backs, as a “dicky” is hung on a coach behind.)
Source: Dictionary of Phrase and Fable, E. Cobham Brewer, 1894