This is in one of the books, maybe The Civil War Musket (?). After you decide which army, when and where (which makes a difference) what I recall off the top of my head is that not every musket was issued with a sling and being part of the accoutrement set they were re-ordered through Ordnance and not regular supplies such as uniform items which went through the Quartermaster Dept. Hence, replacement slings were not automatically issued at any regular interval even if they wore out. Looking at period photos, you can find evidence of muskets with slings and also those without slings.
For the most common imports like the P-53 Enfield, the Federal Government would often order without bayonet scabbards and slings, preferring to issue the US pattern. There are exceptions that can be found in early war photos where US soldiers have English type accoutrement sets and Enfields, but they are relatively rare. The sling for the US model rifle musket merely needs some holes punched higher up and it works well with the Enfield.
Did soldiers re-purpose the musket slings? Yes, there are period accounts in letters and diaries of soldiers using the sling to fasten a blanket roll, or serve as a belt for their trousers.