Become the reason somebody can confidently delegate.

Delegation only works if the person handing it over can stop thinking about it. Otherwise they have added a management task rather than removed a work task, and they will eventually stop delegating to you. Being the person who makes real delegation possible is unusually valuable.

When you take something on, take the worry with it. Report back before you are asked, flag problems early, and close it out properly. That is what makes handing you the next thing easy.