Are there companies that would hire you as an engineer for a team lead position without much prior experience? Or is the only way to transition within the same company?
I haven't had any luck applying to leadership positions without prior experience. Your best bet is finding a fast-growing product: when the team triples, you'll need some new managers. I got to move around 2 months into my IC job, and now my team has grown to have another 3 leadership slots.
But also make sure you're going to be happy if it doesn't work out. It's actually more likely at a larger company with established tracks IMO - hit the bullet points for next level/sideways move, say you think you've done so and that's what you want to do, be recommended for or apply internally for such thing. If you were considering leaving as an alternative plan then you presumably won't mind if that's in a different part of the org anyway.
“Team lead” can mean practically anything, but if you’re talking about an EM, I think most companies would be _extremely_ reluctant to hire one with no prior experience as an EM. A bad EM can completely ruin a team; it’s not something that you want to take any more chances with than you have to.
The smaller the company the greater the chance of that IMO. I had to do it within my company and it was all a matter of luck, timing the right manager etc. Today it wouldn't happen again.
I think it's also not easy to do in a static situation where people have a fixed image of you and your talents in their minds. So basically I think you need to move around - if it's not happening for you in one place you need to find another.