What would be the next roles/avenues to look for, after being staff engineer for a couple of years?
Does moving to a managerial role become easy after being a staff for a while, in terms of lateral/vertical hire and also managing the role once hired?
Yes, the most common path after Staff Engineer is a transition into management.
Upward mobility is, frankly, easier on the management chain than on the IC ladder. Growing as a manager is about increasing the # of people reporting to you, not necessarily needing to keep up with the latest framework/tech changes (which can get especially tiring if you've been coding for 10+ years).
Staff Engineers have a ton of context about the team and the respect of their peers, so their transition to management is graceful.
Here's a relevant question about How to make the decision on Staff IC vs manager roles?
Here's all the possible paths:
When it comes to management, I recommend this: [Taro Top 10] Software Engineer To Engineering Manager (IC -> M Track)
For the less common (but still very exciting) transition to product management: [Taro Top 10] Product Management For Engineers