I have a Sony ECM-674 shotgun mike, and it has a diameter a little smaller that most mics. (13/16") Tryiny to find a smaller mount that I can mount on my DSLR, and have tried the clip type mount; but it is very wobbly.

Is there another shotgun that has this diameter?

With most Sony shotguns you either need a rubber wrap or groumet or as I've done I wrap some gaff tape around it and it's worked fine. BTW the groumet is actually about an inch and a half long. It comes with the Sony shotgun on Sony cameras and fits into the mic clip on the camera which might be different than what you're using. Personally I'm tired of losing those so it's gaff tape for me.

Rycote's Invison shock mount can fit a variety of mics.