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Picoides pubescens
Though the bill looks a tad bit large in photo 1, the rest make it obvious that the bill is much too small for Hairy. Definite Downy here.
It's not a red-cockaded woodpecker...the white stripe above the eye is too extends all the way to the nape while the red-cockaded merely has a white speck behind its eyes with prominent white cheeks...looks like a hairy woodpecker judging from the bill