So, my company just built a new website with a new domain name for this company. They still own the old domain name and it redirects to their current site. Is it worth it to contact all of their backlinkers for the previous domain and ask them to change to the new URL?
As long as the old domain has a good 301 redirect to the new one, it should be fine to leave the old links.
You don't need to get all the back links changed if you do the redirect. However, you'll have to keep the old domain forever just to keep the link juice flowing from the redirect.