Unfortunately, no. You should definitely email support@airtable.com about this, but it won’t likely do any good because everybody has been requesting this for years, and Airtable has never implemented this.
In the meantime, the workaround is to create a formula field as your primary field that contains all the different fields that you would ever want to search on in the linked record field.
You can move your autonumber field to become another field, and that field can become a part of your new formula field.
Alternatively, for an external solution to this problem, you can now use Fillout, which lets you specify as many fields as you want that will be searchable in linked record fields.