Coming back to this several years later, some scholars do believe σσ was used to represent a different quality of sound than σ—at least in Anatolia. From Obrador-Cursach's The Phrygian Language page 58, in the context of an alternation between Ανγδισι, Ανδισσι, and Ανδξι in Greek transcriptions:
Indeed, we cannot ignore that ⟨σσ⟩ is used in other Anatolian areas to represent a sound other than /s/. This is the case of Carian [personal names] arliš and uśoλ, found in Greek inscriptions as Αρλισσις and Υσσωλλος respectively.
Based on this, he cites Gusmani for a claim that the Phrygian name could have had a palatalized [ʃ], especially given the surrounding high front vowels.
Adiego's The Carian Language discusses these personal names in only slightly more depth, but does mention that the Carian š and ś are transcribed medially as σσ, while s is transcribed as σ.
However, Melchert (in the Cambridge Encyclopedia of the World's Ancient Languages) points out that Carian λ reflects a historical geminate *ll, and is always transcribed into Greek with two letters (variously λλ or λδ). Similarly, at least some instances of ś go back to historical *ss. He warns that "The nature of the contrast with […] s remains to be defined", but it seems entirely possible to me that Carian λ, ś, and š were actually longer than l and s.
In this case, counter to Obrador-Cursach, σσ could simply indicate a quantity difference instead of a quality difference. Carian phonology is still not very well understood and I'd be hesitant to pin too much of the theory on it.