Identify your peer managers, key stakeholders, and cross-functional partners before you need something from them. Introducing yourself for the first time in a moment of urgency is the wrong time to be starting that relationship.
Most of what slows a new manager down in month two or three isn't a problem inside their own team, it's a dependency on someone outside it they haven't met yet. By the time you need a favor, a fast answer, or a sign-off, it's too late to be building the relationship from zero.
Identify the peer managers you'll actually depend on
Not every peer manager matters equally. Focus on the two or three whose teams your team hands off to or receives work from most often.
Find the person your predecessor always looped in
There's usually someone, in finance, ops, legal, another team, who got quietly consulted on anything nonstandard. Find out who that was before you hit the same situation cold.
Set up a first conversation with no ask attached
A short "getting to know how our teams work together" conversation, with nothing you need from them yet, is a much easier first meeting than one where you're already requesting something.
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Two versions: one for you, specific names and what you'd actually go to them for, and one for your team, more generic, things like a shared HR inbox or the right Jira board, that they can use without needing your relationships.
Finance
[Name], go-to for budget and headcount questions
HR
[Name], go-to for policy and people questions
Engineering & product
[Name] for front end, [Name] for backend, [Name] for data, [Name] for quick fixes, [Name] for brainstorming
Leadership team
[Name] for brainstorming, [Name] for company-wide rethinking, [Name] for leadership concerns
Ops
[Name], has the answer to almost anything operational
Customer support
[Name], knows the clients and is happy to help