My go-to source for such questions is the Online Etymological Dictionary, aka Etymonline.
Here's what it has to say about shift in the sense of shift dress¹.
"body garment, underclothing," 1590s, originally used alike of men's and women's pieces, probably from shift (n.1), which was commonly used in reference to a change of clothes. In 17c., it began to be used as a euphemism for smock, and was itself displaced, for similar reasons of delicacy, in 19c. by chemise.
That probably makes me suspicious², but it's the best we've got. Etymonline is saying "shift dress" comes from the earlier sense of shift (ibid), to move.
TL;DR: Etymonline speculates that shift dress comes from a metaphor of moving (shifting) into a new set of clothes, as in "changing underwear".
¹ Your OED attestation from 1950s specifically describes a slip as "loose", but this appears to be a newer sense of the word, which may explain the discrepancy between that citation and Etymonline's 1590s. For example, smock, an even older word, did not develop a sense of "loose" until the early 20th C (see below).
² The more interesting thing I took away from these etymologies is that shift was originally a euphemism, displacing smock³, which had started to feel like an indelicately direct way of talking about underwear.
In turn, as the wheels of the euphemism treadmill spun through the centuries, shift was itself displaced by chemise. Nothing like speaking French to mask indelicate details!
³ Now smock has a much more satisfying etymology. It appears to come from older words meaning creep, slip, or press close, as in "a tight garment one creeps or slips into".