There are indeed a miread of ways to send faxes. I use several of them.
We have a brother AIO (all in one) that we fax from. I also sometimes fax from Windows. Both are as easy as printing, just choose a different "printer" and type the address. You can even set your cover letter to say what you want. Not quite sure how your HP AIO works but it is probably similar.

There is updox and internet faxing. I haven't used them

Are cover letters necessary? Depends on how well you know the recipient and whether they are expecting the fax. It is nice to have a telephone and return fax number if you accidently get the wrong fax or not all the pages go through.

Wendell
Pediatrician in Chicago

The patient's expectation is that you have all the answers, sometimes they just don't like the answer you have for them