Hi,
You can add the newly added field to the _rec_names_search of the product.product model.

See: Rec names Search

Thanks

For that you need to add "_name_search" method in your custom module in product.template object.

	​@api.model
	​def _name_search(self, name, domain=None, operator='ilike', limit=None, order=None):
        domain = list(domain or [])
        if name:
            domain += ['|', ('name', operator, name),
                     ('your_custom_field', operator, name)]
        return self._search(domain, limit=limit, order=order)

Have you tried inheriting "name_get" function?

What it does is changing the name of search result.

https://www.cybrosys.com/blog/how-to-use-of-name-get-function-in-odoo