I try to stay away from the gill plate as it carries loads of bacteria. On bigger fish just hook your thumb under the outside edge of the jaw and stick your fingers in his mouth (like you are grabbing a baseball bat). Just be careful of hooks.
If you don't want to lip it you can grab it just behind the head with your thumb and index finger, I try not to touch there gills, it is like sticking your hand in there lungs
If you want to lift it up after grabbing it by the lip, on the bigger fish you will be grabbing the lower lip with your hand, not your thumb. When you swing it up, put a hand under its belly to give it extra support.
"It was the blackest night! There was no moon in sight! (You know the stars ain't shinnin cause the sky's too tight) "