How important is it for you to be connected within your org to move up as an IC?

Alex is spot on! It's super important as an IC to be well-connected, especially within your org, in order to achieve your own goals and the goals of people who depends on you.

This is the masterclass Alex was talking about: https://www.jointaro.com/lesson/tyaPFMXPoz2pmzB7OVzt/how-to-build-deep-relationships-quickly-in-tech-12172022/. The best advice that applies at all times IMO is give credit to / thank others meaningfully.

Since you mentioned "as you moved up the IC levels", I can describe the role of people I believe people should focus their energy on finding and connecting with, for each level that I've experienced thus far. After all, we all have a finite tank of emotional energy to spend - so who we spend it with does matter.

Note, direct manager is not included here because that's an important person to have a connection with no matter the level.

Entry Level (L3): Look for great peers.

1. Look for a rubber-duck buddy, likely someone of slightly more experience than you. Early on in my career, it was super helpful to compare notes with someone whom I felt really comfortable talking to. I wouldn't understand many things, and maybe they wouldn't either, but you could discuss and figure it out together. Or maybe they've spent 2 days debugging the same type of issues you are facing now and you can save two days learning from them. Or maybe you can talk about which senior engineers in your team are actually helpful for specific types of questions. The biggest challenge of entry level engineer is onboarding properly, and having a buddy who's done it recently is a tremendous asset (plus, they can get credit too for helping you!)
2. Look for a mentor who can take you under their wings. This could your TL, your manager, or someone outside your direct team. Having a reliable mentor who's honest with you is great because they can generally give feedback to help you grow without you feeling like they are judging you (compared to say your manager giving you feedback). They can also share parts of the promo process without having to be politically correct. I personally had someone like this when I was at Amazon, who helped me put together my promo doc and ALSO held me accountable privately when they noticed that I was slacking off.

Mid Level (L4): Look for great leads.

1. Look for technical leads, whether or not they have the "lead" title. They are often the thought leaders of the organization and those who contribute greatly to engineering decisions. As your goal as an L4 now is to independently deliver projects, on time, it means you need to be very much be in sync with your leads and support them where possible. As an L4, team leads not only look to you to not be a burden ("be independent"), but also to help them where possible. Spot a gap in your feature or risk that your lead didn't see? Volunteer yourself to address it. Your leads are often distracted by your team's many concurrent projects, but you, as an L4, have more opportunity to focus on just a few projects. This means you will have the time to discover issues that sometimes can only be uncovered during implementation, such as a needed refactor. Do these when possible. In addition to learning a lot, it will also earn you lots of trust and appreciation from your leads. Finally, spotting work for yourself like this is also a great sign you are getting ready for senior level responsibilities.
2. Connect with your project peers. Often, your main projects will span multiple teams and you'll have peers working on a different parts of stack together on the same projects. It's important you don't funnel all discussions through your leads (or processes) but talk directly to your partner engineers, understand their plans, and adjust accordingly. Own your project and make sure it ships - and that includes building alliances with people on the other side of fence.

Senior Level (L5): Look for great partners. And build up your own team.

I'm still working on this so I'll link this talk that I found extremely helpful: Good vs Great Tech Leads. But in-essence:

1. Build great relationships with cross-functional partners. This can be PM, TPM, EMs, or UX. Engineers at this level will be having fundamentally different types of discussion than L4 (not just "how do we build this", but also "what should we build"). As such, it's super important to have regular discussions and have great working relationships with your cross functional partners, so that you don't discover that your team has just spent a year building something that no one in the outside world wanted.
2. Find people you can up-level and build yourself an informal team whom you can rely on. While it's important at lower levels to mentor junior peers or new members of the team, this isn't a necessity - but at senior level this becomes almost part of the expectation. As a tech lead myself, I find that I can add the most value to my team's own delivery by coaching the engineers on my team on the mindsets and skills necessary to be great contributors. They in turn do great work so that I don't have worry as much. You don't even need to know all the answers. Sometimes it's just telling them who in the organization they should talk to or how to act when an emergency occurs. Be the rising tide that lift all boats - or if you prefer a football reference - A great QB elevates everyone else around them.

I'm not qualified to speak on L6+. But I heard https://staffeng.com/book is a great resource for this. Hope you found this answer helpful! Good luck on your journey.