Passing default values when adding to cart on the shop page

Many thanks for the excellent plugin.

I’d like a frontend text area field as a note for all products in an order. This works great when adding to the cart from the single product page as the field a default value can be passed even if the customer does not edit the field. These then display and can be edited in the cart page.

However, for products added to the cart from the shop page or any other archive, this isn’t possible.

How could I load the field for each product on an archive page? This could then be hidden but the default value still added to the order so it appears with the product item in the cart table?

I’d be very grateful for any pointers.