Is there a way to display a "notify me" button for out-of-stock items instead of hiding them on Shopify?

Unless you want to change some code, the best way for your customers to receive notifications when products are back in stock is by adding a Shopify App that was created for such a purpose.

Apps like Back in Stock, Pre-Order Manager, and Restock are all designed to send back-in-stock notifications to your customers.

Simply install your desired app, head to your product page, select the product you wish to add the notification button to and find the “Notify me when this product is available” button. This may look slightly different, depending on the app you use, but it's usually located under the “Sold out button“.