I was able to easily play a standard G chord, as in the first example of the excellent video in the link above, by slightly rotating my left hand upward, toward the headstock, which for a split second, in the "crook" of my hand and thumb. It allows those 3 fingers to be placed on strings 2 3 and 4 more horizontally, making it easier to finger them properly without crowding. At least, that's what worked for me. It also hasn't affected my ability to move to the next chord either.
PS: Same technique to facilitate the E also.