Yeah, I really like that "embiggen" as well....
The trick is that you want to make the sides and back strap fuller while not making the front strap fuller as well. The idea is to embiggen the grip area towards the rear so you can obtain a better sort of reach to the trigger.
Hmmmmm....... The first though that comes to mind is custom wood grips made up by a good wood worker or wood savvy gunsmith. But you don't even know yet how much embiggening (oh look! Now it's an adverb as well!) you need, right?
One option for the time being would be some art store foam or perhaps some bicycle road bike foam rubber "cork" as used on handlebars for road bikes. The bicycle foam tape might be best since it is already tapered so it lays and builds smoothly.
The trick will be that you want to use it vertically since you don't want to build the front up in thickness, just the sides and rear. But it doesn't stay in place all that well by itself since it has some whimpy adhesive tape in a thin band along the center. But if you apply it to build the thickness and width then wrap it LIGHTLY with the fabric hockey tape you'll at least get an idea of what sort of size works for you.
Once you can come up with some dimensions you can then look into getting something more permanent made up. This could be wood grips or molded synthetic rubber composition. Then you won't get tape goo all over your hands.
In the meantime may I suggest a 2011 style gun or a Beretta 92fs? Other than the Desert Eagle, which is just plain and simply a mutant, these guns have the biggest and fattest grip size I've ever tried. It sounds like you'd still need more. But these would at least get you started.