If becoming a senior involves expanding scope/finding your own scope at Big Tech, how does this work alongside a product manager?

The culture will vary from company to company, but in general, I recommend that you take the time to get to know your PM and how they think if you're a software engineer. At the end of the day, you two are a team: They define the vision, and you, alongside the other software engineers, are the primary executor of that vision.

Here are some resources that concretely talk about how this collaboration can manifest:

• System Design Series: Taro Playlists [Part 3] - Engineers should go back-and-forth with the PM to make sure that the product requirements are granular and crystal clear. Since you're a mid-level engineer, I highly recommend just watching my entire system design series as well.
• "How to converse with a non technical PM?" - This Q&A (from another Booking engineer) talks about how you can really supercharge the partnership between you and your PM, particularly around having them understand engineering feasibility.

Is this a case of just getting them on board with what you have found or just trying to find time to take the initiative?

I'm a bit confused by this question, as I feel like these 2 are usually the same thing. When it comes to creating scope, the process is generally like this.

1. You find time outside of the tickets assigned to you to scope out some potential new project.
2. You convince your team that this new scope you have found is worth pursuing, and the product manager will be a primary person to win over in this discussion. For more advice on how to do this, I recommend this discussion on how to lead without authority.
3. Your new project gets added to the roadmap, and congratulations, you have now made great progress developing senior engineer/tech lead behavior.

The PM will generally be the one in charge of what we develop over the upcoming period (in my experience anyway).

That's true, but this doesn't mean that you can't have a big influence over their vision and tactical roadmap. A common trait I've seen among software engineers who struggle with growth is they box themselves into silos. They don't want to help figure out the vision with the PM. They don't want to fine-tune the mocks with the designers. They just wait for the well-scoped out work to fall onto their plate, and then they code it out. The best software engineers are versatile - They understand a decent amount of several slices of the pie and are able to add value to other people in their organization besides just engineers.

All that being said, how do you forge that much stronger and more productive relationship with your PM? On top of talking to them 1 on 1 every once in a while (maybe even on a recurring basis), I recommend looking into some of the ideas in this discussion about SWE/PM hybridization.