What are some things I can do in order to distinguish myself for a waiver from E-3 to E-4?

Volunteer for work. Be the go to guy, the reliable person. Take charge in the absence of leaders.

SoM boards, PT scores, Correspondence courses

Own your job, take pride in every facet of it, no matter how small. Show improvement (PT, Correspondence Classes). Own your uniform and bearing. Volunteer to lead PRT drills, or Hip pocket training during STT. All this to say, show your leadership you bought into the Army, and you want to leave your mark on it.