So far as I could tell, the new SNP leader has become Leader of the Council, and the administration is basically the same as before all this started: SNP, For Argyll and the Argyll and Bute Independent Group. (Other varieties of Independent are available)
So Argyll and Bute council is effectively run by the SNP's National Council - well, that is the line I would use if I was in opposition to them there.
“If the brain were so simple we could understand it, we would be so simple we couldn't.” (Lyall Watson)
I missed out Mark Findlater, Aberdeenshire, Troup ward, elected as Independent, served in Conservative led coalition, now Conservative.
It might be worth noting in the passing that Cllr Jim Gifford, Conservative, former leader of the council until his coalition suffered the loss of two formerly Lib Dem Independents and two labour, and with it control of the council to a SNP led coalition, not only remains leader of the Conservatives, but in fact remains leader of the Alliance, an opposition coalition of Conservatives, Lib Dems and Independents. not sure it is all that common for a guy to lose power and not only be retained by his own party but by a multi party coalition administration in waiting.