The clarinet is Bobby Holmes and the alto sax is Glyn Paque. I believe both the open and muted solos are by Oliver himself. The muted sound is the big mystery on this one. I think it's an old style Harmon type mute or an ancestor of the Harmon mute, as the closest sound I can get to it is the Emo plastic wow-wow mute, which is like a harmon but has a straight tube instead of a cupped plunger shape out the end of it. Whatever he used, it's a really beautiful sound. Nelson uses the same mute on some of his own records also.
I've always loved this record. Trying to learn the trumpet parts at the moment actually.
Are you familiar with both takes? Take-1 surfaced a few years ago and was released on a 1992 Bluebird album.
Oliver was really on-form on this session (he barely played at all on the previous session) - the other 2 pieces are "Struggle Buggy" and "Don't You Think I Love You", which has a great open solo by him.