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Cold Weather Starting problem

I have a 2001 Chevy Silverado 4wheel drive 6.0 liter engine. 30,000 miles. I live in Wyoming. This problem only happens when it’s below zero. Truck will start just fine on a very cold morning the first time I try it.But if I drive it for awhile then park it for a few hrs. it will not start again. The lights will come on, the instrument panel lights up but the starter will not engage, won’t even click. When the temp warms up it will start like normal. I have checked the starter cable connections, they seem to be tight but I haven’t actually removed and cleaned them. I have cleaned and retighened the battery cables. Could this be a moisture problem somewhere in the starting system?