I agree with Pat.
A different "set" of microbes decomposes woody materials (cellulose/lignin) than decompose leafy materials, but IF you regularly have green material (such as freshly-aged grass clippings or fresh cow manure) on hand that can be added to the 'sawdust mix' as it is unloaded, then that's a whole different situation - Your goal is to get the carbon/nitrogen level at a point where your pile is HOT.
Compost piles will heat up if you keep the nitrogen-carbon ratio so that you have a good supply of rich high nitrogen and high protein materials mixed in your compost pile.
You must have at least 2-3 times more carbon matrerials than nitrogen materials in the pile also. The carbon will give you odor control, and also you can't produce humates in the soil without carbon materials.
You must keep the pile constantly aerated and moisturized. You don't have to turn a compost pile, but you can aerate it by poking it with a broom or stick, or using stationary aeration pipes or vents in the pile.
Make sure to manage the compost pile so that it does not have any funky smells, and keep the carbon materials on top of the pile for odor control.
I heat up my large compost stockpiles by using high nitrogen/protein, dry molasses, compost/manure teas as nitrogen/microbial activators. This way I add water, microbes, sugars, and nitrogen to my compost piles every time I use it. The microbes grow like crazy, and the pile gets hot very quickly.
Your answers are relative to adjusting the 'regular' pile 1st-heat temperature properly, which might indicate addition of a sugar/starch amendment, since pile heat is mainly caused by thermophyllic fungi 'eating' the initial available sugars and carbohydrates from the material you build the pile with. High-carbon materials (Browns) are usually deficient in sugars/starches in a form that heat-producing fungi can assimilate.
You can incorporate an alternating layer of a very high-carbon mix into a 'regular' pile as you would an alternate layer of dry leaves - but how much, depends also on what your 'regular' carbon/nigrogen pile materials are.
Second focus would be on what additional material would need to be added during the 1st turn, to 'round out' the pile material to obtain a 2nd heat, and to begin maximizing production of humus percentage