I cover my wood depending on its importance
Just because.
Every time it rains lets say 1 or 3 inches then it gets humid I saw mold starting to grow after a few days.
That tells me the wood is wet.
I have stored wood in the small tractor shed
On that pile I had time to play with so when I was splitting, I debarked the Ash, it dried but not as good as the pile in the sun and wind, I will not store wood in the small tractor shed to dry again.
Then there is this pile the first pile it is never covered the wood there was a nice brown color now its grey and has black spots (mold)
Then there are these three piles, my main piles, I save these for the coldest part of winter, you know when the power goes out and its below 0 maybe 20 to 40 or so below and there is a blizzard, I always cover this pile making sure the sides flap in the wind this removes the moisture and the black plastic attracts the suns heat, I do uncover when there is no threat of rain.
With this I get a good dry wood that when split there is barely any trace of smell after about 3 months.
I start getting nervous this time of year because its ready to go and the night time due gets into the wood if uncovered I can smell the wood again.
This is uncovered and next years wood, it will be ready by spring, the dry snow pack of the winter will dry the wood it needs to be covered before the spring snows.
(note the big Elm)
I was unable to split so I practice boring with the tip of the bar this will split easy next year
My bottom line,
wood is the heat I use the most, gas and electric only go on to keep the pipes from freezing when there is no snow to cover the bottom of the house.
I take wood very serous, and to cover or not to cover depends on the importance of the wood.