web hosting – Search engines indexing only the old site which has been moved to a subdomain

I recently took up a project for a website overhaul. I went on to deploy my website on the client’s hosting service. It uses cPanel. On the hosting service they had the old version of the website.

I moved that to a subdomain named old.example.com. I rigged my version of the website to the main domain. Everything seems to work except for one thing.

If I search the web for their website, the only search result that appears is the old.example.com. The only way to access the current version is via direct link.

The old version of the website was maintained by another outside developer.

Is it possible the old developer is running some sort of indexing service or what not on their version that automatically updates the domain of their site or is this some sort of cPanel option that I can change?

Forgive any vagueness in the question, if there is more info that I should provide, I’ll add it in an edit. It’s also important to note that the clients aren’t all too tech savvy so the amount of info I can acquire is quite limited.