Looking to lateral to NYC and seeking advice

I'm currently an An1 at a regional boutique in the Atlanta area and am hoping to lateral to a bigger, more well-known bank in NYC but have a bunch of questions. Hoping I can get some advice / assistance.

Here's a little background info. The bank that I'm currently at does not have an office in NY. Despite the overall decline in M&A, we deal primarily in the LMM space and run pretty lean teams so I've had a good amount of experience. I went to a non-target but decent school (think Wake Forest, Tulane, TCU) meaning that I don't have a ton of natural connections up north. I am registered with FINRA but I don't have all of my licenses yet. I'll have those completed by the time I formally begin this process. I think my goal is to be in NY by Jan 1 25 so I'm just trying to get ahead of the 8-ball.

Is there a set time when most people bounce to PE / other firms / out of the industry and An2 roles open up? Like a busy season for this type of recruitment? Is it possible to get promoted to an An2 while also lateraling in or would I have to re-do my An1 year? Is it realistically possible to lateral into a bulge or should I stay in my world of boutiques? Given my limited connections, I'm thinking I should reach out to some headhunters so does anyone have a good recommendation IB laterals? Should I be cold messaging / emailing people on top of a HH? How long would the whole lateral recruitment process take? Like from submitting an application to start date? Although I'm fine spending my non-working hours applying, I'd imagine that all interviews will be conducted during normal business hours so how do I work around that given that I go in every day. Do people take PTO for interviews, meeting with headhunters, etc? Anything else that I should consider or tips from someone who has done this before?

Thank you, I really appreciate the help.