Keep in mind there is no right or wrong, just what works for you.
Sometimes I literally put my left hand in my pocket when doing picking exercises because I am trying to focus all my attention to what my right hand is doing. Other times, I hold 6 string barre chords to develop my grip and left thumb strength but there are other exercises I use for doing that sort of thing. Personally, I prefer the E major chord...it just sounds awesome on the guitar.
After a while, you NEED to change chords with your picking exercises because the changing string tension will throw your right hand into uncharted waters and it sounds WAY more musical.