How to apply for FT roles if I already have an FT offer #London

So, I’ve got an offer for an off cycle internship that is convertible with the immediate start after the internship. Basically, you do 6 months and then go on to join for full-time in September 2024.

The firm is very nice and very reputable, an EB (let’s name it ABC for this post) but I’d like to try other places due to my interest in another sector.

I was planning initially to start applying in September 2024 for other graduate programs whilst doing full-time at this firm (provided I get an offer), but I’m slightly concerned because regular grad programs start at AN1 and by the time I join another firm - I already would have completed AN1 at ABC. I have no problem starting as AN1 again because the firm I’m targeting one is the prominent BB I have some good network at but I am concerned would this BB or other firms even consider me for regular grad program that you apply for online if I’ll already be doing AN1 at ABC?

I have a network at this BB firm and was thinking to get reference and ask them to push my CV a bit for HR, but it’s a bit shaky to just expect HR to reach out to you and it’s equally shaky to just wait for random role for AN2 to open up?

I’m seeking some guidance, should I take my shot and apply to regular grad online as per usual and hope for the best or wait for AN2 position to open up and apply then with the references from seniors?

Thanks!