The circumflex of "Saône" seems to have been intended to reflect the pronunciation (like that of Rhône) and isn't etymological. In Middle French, the spelling was in concurrence with Sonne and Saulne, the latter of which seems to be another attempt to reflect the pronunciation /'sɔ:.nə/ (modern /so:n/).
Which leads us to the main issue with this question: since circumflexes originated as a scribal device to indicate an unwritten letter, and lost consonants tended to cause compensatory lengthening, the presence of a circumflex in the modern spelling convention might be motivated by the etymology or the phonology of the word... or both. A non-etymological consonant (and especially ⟨s⟩ and ⟨l⟩) was also used indicate a long vowel, as we saw with ⟨Saulne⟩
In the case of "dîme" for example, the historical ⟨x⟩ was probably used to indicate a long vowel (while hinting at the Latin etymon) and was in competition with ⟨disme⟩. So is the modern ⟨î⟩ replacing this ⟨x⟩, or simply another convention used to represent the same phoneme? Look at the river Aisne, with a similar starting point (Axonam), but whose spelling was never updated to ⟨Aîne⟩.
Likewise "âme", from animam, which appears in OF with variants such as ⟨anme⟩, ⟨alme⟩ or ⟨arme⟩, that might reflect either the actual pronunciation of a consonant, or just represent a long vowel. The word eventually settles on the spellings ⟨ame⟩ and ⟨âme⟩, but should we consider the modern circumflex a replacement of those old letters, or just an alternate spelling of the same phonological phenomenon?