A photograph of the “Military telegraph construction corps” taken by Alexander Gardner in April 1865:
You can read more about this photograph at the Cornell Library. Apparently as the Overland campaign began in May 1864 a new insulated wire began being used by the telegraph corp. The wire could be placed on a mule’s back and run off quite easily. Earlier wire had to be dispensed from a wagon.
Read a lot more about the Union telegraph work at civilwarsignals