SEO Advice For Established Authority Site

Greetings ! I have a friend that has a pretty big authority website on the travel niche in my country and is currently ranking #6 for a pretty big and competitive keyword. He has a few backlinks, around 6k coming from 39 domains (They are not spam at all, the reason for the high number is due to a partner site placing his link on their blog roll. )
From my analysis, the reason for his current rank is due to the vast amount of unique high quality content he has published in the last 4 years.
He recently asked for my help to build him a new site because his old site looked outdated and was not responsive at all. I built him the website and also teached him a few on page seo stuff.
I happened to have a pretty good domain with TF 32 and a good amount of links pointed, on the same niche as him and I made a blogpost pointing to his site.
So my question is what else can I do to maybe push his rankings for this main keyword he is ranking 6th to 1st spot or maybe top 3 ? Any tips ? Buy Pbn links ? Can I be more aggressive ?

Thank you

Have you tried building Schema?

so from now you can try to build some edu and gov link building for improve da/pa metric because in top ten rank pa/da also play big role.

Have you analyzed competitors? get backlinks from the authority sites they have their backlinks on.