That's even easier to fix because the muting is built into the musical phrase you are trying to do.

Originally Posted by: ken.needham

I'm practicing moving from one fret to the next on the same String (High E). As I pluck the string on the first fret and lift my finger . . .

If you just playing a note on the same string leave your index finger in place, just slightly release pressure, but maintain contact with the string. The only time any string should be free to ring is when you want it to sound a note. Otherwise you need to mute the strings you don't want sounding.

Make sense?