Well I have been having problems with C2 ,or long C as I've heard it called, ever since I took up the clarinet. Most of the time when I went from a low register note to C2 I would usually squeek or the note just wouldn't pop out. For the life of me I could never figure out what the problem was. Took the clarinet to the tech and he could find nothing nothing wrong with the instrument. finally after taking a real close look at my finger position when finging for the C2, I noticed the rubber thumb rest that has been on my clarinet since I bought it. I then realized that the thumb rest was positioning my thumb so far off the thumb rest that my right index finger didn't quite cover the hole when fingering for C2. I removed the thumb rest and I think that cured about 90% of the problem.

Below so that the clarinet would rest on my thumb.

i'm not sure what note you mean by c2.. the c in the staff, below, or above?

anyway, cut up rubber pencil grips make nice thumbrests. durrring marching band my freshman year, a girl got a pack of them, cut them up, and gave them out to us all. it was excellent, because if (when) they inevitably fell off, she had a bajillion more. plus, people with the same model clarinet could use different colored rests to differentiate easily.

You can also use about an inch of surgical tubing. You can buy it at any medical supply shop. Make sure you get the stuff that's a quarter inch thick. The advantage of this set-up is that is allows you to slide your thumb further to the left, while maintaining stability.