Handle the small things well and people start handing you bigger ones.

Nobody hands over something important based on a promise. They hand it over based on evidence, and the only evidence available early is how you dealt with something small. Returning the equipment, answering the message, finishing the boring half of a task: each one is a data point, and enough of them change what you get offered.

Look at the least glamorous responsibility you currently have and do it noticeably well for two weeks. You are not proving you are impressive. You are giving somebody a reason to risk something on you.