In the first case, you may have worked in your department so long you can no longer “see” things differently because of limited experiences or you feel caught in the trap of day-to-day routines. In this case, try some mind exercises before beginning an improvement effort. For some ideas try these ideas.
In the second case, think about the handoffs your department makes to other departments and talk with them about how to improve the handoff. Most colleagues want to do a good job and, as long as they look good to their management, they will happily work with you to improve the process. Remember to sell “what’s in it for them.”