Congratulations on your deer. You may be overthinking things. Field dressing, so long as you get the organs, intestines, man-parts and any poop, goop or foreign matter out, it's a good job. Tidiness will come with experience. I do not generally quarter in the field unless dragging/carting to a vehicle is not a viable option. I use a commercial deer cart, the cheap $50-70 kind and have taken them out of up to a mile of broken country with this. Keep about 20' of paracord or similar rope wrapped around it ready to go, the provided tie downs suck and you'll want more. In situations where packing and quartering may be necessary, I carry an ALICE military frame pack and lots of cable pulling line (paracord will work well). The external frame has plenty of tie down points if anything won't fit inside like hide and antlers. A couple of 30 mil contractor bags keep everything clean.
I usually skin and quarter just prior to butchering. A poly picnic table is the butcher table, and a couple of disposable construction boards work well for cutting on. 2x12 scraps are perfect. I tape a layer of butcher's paper to the table, and throw down some poly drop cloth, blood is a pain to clean from textured concrete. Hang the deer from the hind leg tendons with a skinning gambrel (light duty "deer" ones are inexpensive and work well enough). You'll want to add a swivel and carabiner, and a block and tackle or at least a pully will make the chore easier. Skin and quarter and bring the quarters directly to the butcher table and things will be cleaner and your venison won't dry out. Use duct tape to remove the majority of the hair, a propane torch to rid the meat of any stragglers. Hope this helps with your next one!
Edit...read one of your follow up posts on the reason for quartering in the field. In that situation, you can buy some time by placing one or 2 blocks or bags of ice in the cavity and covering the deer in the shade while you rig proper gear for hanging and quartering. Our kit in warmer weather includes a couple of milk jugs frozen full of water. Keeps the beer and deer cold if needed. Had to do a hasty butcher in camp some years ago. Ran a rope over a heavy limb on a tree, and used the pickup to hoist the deer by the hind legs with a homemade gambrel (2x6 section with a lag bolt in either end). Used a folding poly picnic table to butcher on. Much easier than skinning or quartering on a tailgate.