Add fields to search in the product search form

Hi,
I'd like to know if it's possible to add or customize the fields available to search in the product search form. In my case, since I'm planning a migration from an old software, I need to include product information that doesn't fit in the current structure. Information like store and logistics references only for internal use.

Those have to be available for searching. My initial plan was to concatenate those into the Tags field but it seems to be missing from the available fields to search in.

Alternatively I would like to manually add some fields in the prestashop product table to store that extra information, so would it be possible to add those to the list to search in? or at least include existing ones like tags or different languages fields?