So where does this version come from? It's not new (1860s according to dictionary.com). There's speculation at wordreference.com that it's a regional variation, but nothing conclusive.
A number of sources agree on the date in which it was apparently first used, that is 1862.
Reach-me-down "ready-made" (of clothes) is recorded from 1862, from notion of being on the rack in a finished state.
From the Routledge Dictionary of Historical Slang and the Dictionary of Slang and unconventional English by Eric Patridge:
adj. 1862 (Thackeray; Besant & Rice). Ready-made, or occ. second-hand, clothes; in late C.19–20, often of such, hence of any, trousers: perhaps always S.E. (ex U.S.). Coll. reacher.
It appears that hand-me-down has a later origin:
Something, esp clothing, used by one person and then passed to another, esp to a younger sibling : I wore mostly my brother's hand-me-downs (1874+)
(The Dictionary of American Slang, Fourth Edition)
OED offers as its earliest attestation of the adjective 'reach-me-down', in the sense of "ready-made", an 1861 quote from Thackeray's The Adventures of Philip on His Way Through the World, as published in the November issue of Cornhill Magazine.
You know in the Palias Royal they hang out the most splendid reach-me-down dressing-gowns, waistcoats, and so forth. "No," thought Philip..."My brown velvet dress waistcoat...is a much more tasty thing than these gaudy ready-made articles."
The adjective, however, was well-known by 1861, and shows up in print at least as early as 1844:
His "smalls" were white...they looked as if they had not been "made to measure," but had more of the cut of the Blue-coat School, or a "reach-me-down shop."
Significance should not be attached to the first appearance I found being in a US publication: the article's author is Lord William Lennox, and it deals with English horse racing.
Of the adjective, OED notes "Also in later use: cast-off, second hand." The finding that the sense of 'cast-off' appeared later corresponded with my own, and suggests that 'reach-me-down' did not originate as a regional variant of 'hand-me-down'.
OED attests noun usage of 'reach-me-down' from 1861, but the earliest use as a noun I found was from 1849:
...the Dean, who was never very particular in his dress, encased himself in a very ill-shaped, ready-made-looking, great-coat, a regular "reach-me-down" from a slop shop in St. Mary Axe, ....
Of the use as a noun, OED again notes "In later-use esp.: an article of second hand clothing; a hand-me-down."
About 'hand-me-down', OED attests the adjective from 1826 and notes "Occasionally also: ready made, as opposed to tailored; off-the-peg." So too is the OED note concerning noun use of 'hand-me-down' (1835) weakened: "(also) a ready-made garment (now rare)."
Precative or jussive use of the older verbal phrases 'hand me down' and 'reach me down' may be synonymous, that is,
Hand me down that suit [from the clothes rail in a shop].
may be used as the semantic equivalent of the colloquial
Reach me down that suit [etc.].
and vice versa. The possibility of such synonymous use of the verbal phrases explains the "occasional" (OED) noun, and "now rare" (OED) adjectival, uses of 'hand-me-down' in the sense of "ready-made" more usually associated with 'reach-me-down'. Equally, that possibility of synonymy explains the opposite, wherein the colloquial 'reach-me-down' is used to mean 'hand-me-down' in the sense of "[something] passed on".
Contributing to the weak tendency to use 'hand-me-down' to mean 'reach-me-down', and vice versa, is a conspicuous empirical association: both "passed on" and "ready made" clothes often fit poorly.
The possible synonymy of the two terms, and their conspicuous empirical correspondence, are where "'reach-me-down' (as a variant of 'hand-me-down') comes from".