Careful what you kill and who catches you doing it. Some woodpeckers are protected in TN and carry a $600 fine for killing just one. Tree huggers will turn your a...ss in quicker than you can say Jack.
Woodpeckers are territorial and when they move in, they're hard to get rid of. I know a guy who had a house that the woodpeckers attacked. He tried many repellents including fake owls, metallic streamers, phony snakes, vinegar washes, etc. All yielded poor results. He lived in a neighborhood where guns were out of the question and his next door neighbors were tree huggers. He knew they would report him if he killed one of their beloved birds. Then he stumbled onto a very effective solution. He bought a few small mouse traps (the woodpeckers were the small variety.) He drilled a hole in the mousetraps so he could hang them on the house, baited them with suet, and hung them on small finishing nails (no head to force thru the small hole).
Just as he expected, his tree hugging neighbors saw the mouse traps and asked their purpose. He told them they were there to scare the squirrels away - they had invaded his attic. He assured them the small traps were painful enough to teach them to keep away, but did not permanently damage them. The neighbors believed the yarn and went on their merry way.
The last time I talked to him, he said he had caught 12 and reported that he no longer keeps the traps set all the time. Now, he only sets them when he hears a new peck. He did say that he had to be very careful to keep the tree huggers from seeing what he caught. At times, he could not set them if he was going to be away that day. He had to check the traps every morning and afternoon when they were set; otherwise, the tree huggers might see what he caught.
A man's gotta do what he's gotta do.