I just got promoted to senior engineer level (E5) at Google and I just switched into an infra team 2 months ago. In order to succeed as a senior engineer, I know I need to mentor and influence other engineers, specifically helping newer and more junior folks on the team.
How can I do that quickly, acknowledging that I’m not their direct manager?
The most important (and smoothest) is to set the example. Strive to write clean, non-hacky code. Leave thoughtful comments on code review. Be kind while communicating clearly and firmly. Passion and positive energy are infectious when high enough - Try to "overwhelm" your more junior teammates with it.
If you want to go deeper, you can always just mentor them directly. There's nothing stopping you from striking up a 1:1 series with the junior SWEs on your team who you think are considerate and approachable. This is great for your manager as well since it gives these reports another source of support and mentorship, lessening them as a SPOF.