A “jacket” is something one is known for, like a reputation, but often with negative connotations. One common phrase is “snitch jacket,” which means that a person has a reputation for providing information to prison officials. We recently had someone discuss having a friend who is trans as giving him a jacket that brought about abuse for being friendly or affiliated with LGBTQ persons.
A jacket can follow you from unit to unit via prison communications networks. Your jacket can mean that you have problems at a new unit almost as soon as you arrive.