Need advice on moving up.

I’ve only been here for about 2.5 months, but feel like I’ve already grown to be one of the most responsive and productive associates. Pretty much every TL calls on me to train or to do things constantly. When I was having a hard day and ranting (leaving out a lot of details here for space-sake lol) to a TL she explained that I was one of the more trustworthy associates. Generally OGP here is a shitshow, boatloads of favoritism, elitism, and it’s my goal to rise (whether over months, years, or perhaps never) to become a TL and try to get things right. Should I just keep doing what I’m doing and let my coach know that I’m interested in moving up and just see if it ever happens? Any advice is appreciated. I don’t seek the power, but I really do want what’s best for our OGP team in whatever way I can.