Trying to do a search within a search result. Don't know what the proper term for it is. Looked around and realized it is done very unApple like. In the search box I have to actually type something like from:sender_name subject:payment. Is this right? Isn't there a better way? I don't think it is even documented.

In Mail.app? When you begin a search, it gives you some options to limit the search ("from" etc) or how to sort (by date, sender, etc). Other than that, can't help!

---

But can you search inside a search?

You can't do a search inside a search AFAIK but mail is usually pretty good at searching for both -- e.g., using your example, in the search box type "sender_name payment"

In my quick test, searching for a common subject for me gave 270 results, adding a sender name to the search knocked it down to 2 (the original email and my reply).