My manager has 3 pods reporting to them, and my pod is pretty small at just a couple engineers. Because of this, I'm not entirely sure what I can take on to start making progress towards Staff. What projects can I work to fill in this gap? For additional context, I work on the back-end, but I can do full-stack if needed.
At a higher/abstract level, you should always take a step back to reflect what you can do to make the company more successful. What are the things you think you can contribute/lead to help improve the product, the process and the productivity of the team. In an ideal world, promotion is an after thought. Focus on learning and making the product better. Promotion will come naturally.
However, we do not live in an ideal world, so let's move onto the more tactical level. The paths to Staff engineers can vary. However staff engineers in general are expected to excel in following things
Based on those expectations, you can reflect upon what are the gaps in your current behavior. You should also work with managers and ask him/her to help closing those gaps.
Specifically to your question, a small pod in a startup does not prevent you from finding the path to grow to staff. On the contrary, it gives you the opportunity to demonstrate
To be a staff, it's no about grabbing a big project for yourself and optimize for the maximum impact possible. It's about becoming the 10x engineers that make the team, org and the company function better.
Good luck :)