Hasselbach 2005 discusses the use of ś in OAkk transcription, which she says was introduced by von Soden, but ultimately rejects it as unclear.
According to her, modern (post-Gelb) scholars generally reconstruct three sibilant "phonemes" for Sargonic Akkadian, which are consistently written differently. There may have actually been more phonemic contrasts here, but if so they were hidden by the writing system.
In von Soden's transcription:
- ś (Hasselbach's s) comes from Proto-Semitic *š and *ś, and is written with the Sumerian S signs
- š (Hasselbach's θ) comes from Proto-Semitic *θ, and is written with the Sumerian Š signs
- s (Hasselbach's z) comes from Proto-Semitic *s, *z, *s', *ð, *θ', and ś', and is written with the Sumerian Z signs
In other words, ś represents the reflex of Proto-Semitic *š and *ś when written with the S signs in Sargonic Akkadian. In Classical Akkadian, the reflex of these two PS consonants was written with the Š signs, and is thus transcribed as š instead.
Even von Soden later seems to have regretted this particular convention, writing in 1995 (my translation):
In other words, the Old Akkadian phoneme ś is a phantom, originating in a confusion between graphemes and phonemes. There are probably no sibilants other than those in Old Babylonian; they are just divided up and written differently.