I'm working as a manager and transitioned to this role for the same reason you mentioned. However, I had been talking to my LM for some time, and he found an opportunity for me.
Simplest way for you to get one more mentee and form a team of 3. The next step is to add one more person to the team. Once your current team lead receives a promotion, you will be next in line for their role. That's the path I'd follow.
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I built a game about being a manager at FAANG - https://devmanager.carrd.co/
I'm working as a manager and transitioned to this role for the same reason you mentioned. However, I had been talking to my LM for some time, and he found an opportunity for me.
Simplest way for you to get one more mentee and form a team of 3. The next step is to add one more person to the team. Once your current team lead receives a promotion, you will be next in line for their role. That's the path I'd follow.