Show that you can handle the pressure
If you fluster about stuff that doesn’t matter, do you really think your boss would trust that you could handle additional work/pressure? No. If you’re doing well, make it obvious that you’re completely capable of managing your workload and/or others’ at times. You need to be able to handle everything that’s thrown at you, regardless of the stress it may temporarily cause. Don’t get flustered by by the small things in life if you want to have big things in the future - you can use that philosophy for pretty much everything (you’re welcome).
Do your job ridiculously well
So this is tip is to prevent you from doing the opposite of flustering. If you want to climb your way to the top, you need to give a crap about the work you produce. Don’t just get by with the expected day-to-day doings, make a bloody effort and prove you’re ready for the next stage. Even if you can’t do everything, the “I don’t have that answer for you yet, but I’m working on it,” response will go a long way in proving your dedication.
Volunteer whenever the opportunity strikes
That charity event the company’s holding? Offer to help out for a few hours. The project you know about, that’s falling way below expectations? Give a helping hand. Helping out when you’re not expected to, makes it clear that you’re here to genuinely help the company grow, and not just because you have to show up every day.