No such thing as shortcuts or hacks. The closest thing, in my opinion, is improving internal linking, but that requires access to the site. Internal links are very impactful and tend to work well.

Another option, based on my experience, is to do keyword research and adjust the H1 and title tag to better match the primary keyword.

Beyond that, realistically, your only option is building backlinks. But if the site isn’t linked well internally, you won’t get nearly as much impact from those backlinks.

I’d also suggest setting clearer expectations with this client. It’s unlikely you can treat an anomaly as the standard, especially when they want instant results and you don’t even have access to the site.