I agree that the mic on the logitec is not the best for music. I use and Audio technica 2020 USB and it is perfect. The web cam recognizes it as my primary mic so i don't have to reconfigure it my settings every time. In fact the volume setting also remains right where I need it.
A sure way to improve the sound with a mic of your choice is to connect it to a USB Audio Interface, like the Focusrite Scarlett 2i2, which is about as inexpensive as they come. You can hook a dynamic or a condenser mic (the audio interface usually already has +48V phantom power available).