Background check - applying during notice period

Hi,

Long story short. I have worked in IBD for a while and due to bad cultural fit and an aspiration to lateral I decided to quit a while back (while having another job lined up, altough not signed).

I did enter another process for a job a really want (not the first job lined up) and recently found out I got this second job. Contract to be signed any day and I am obviously very excited. My previous job is an ongoing employment on my CV (since I was still employed during notice period when I applied). My last day was last week.

How should I handle this in terms of a potential background check? Advice would be greatly appreciated.