Hi Ellen,
You can edit the "/YOUR-ROOT/themes/YOUR-THEME/templates/catalog/product.tpl" file and put the code where you want it.
Alternately you can use our module below to show the date and also manage the stock notifications. See the demo below!
https://prestaspot.com/home-accessories/10-24-brown-bear-cushion.html#/8-color-white
MODULE: https://addons.prestashop.com/en/emails-notifications/18329-out-of-stock-notification.html