Unity UI Best ways to keep my Canvas between scenes?

This is only my second project where I'm using Unity's UI system; I used to use the IMGUI well past its expiration date so I could control my UI through code.

I'm trying to figure out how I should handle keeping a uniform UI across different scenes, since now my UI and HUD are built on a canvas that exists as part of my scene. I can build a nice HUD in one scene, but how do I have that same HUD in the next scene?

In my last project I kept the HUD in a separate scene with my player's gameobject, and used scene manager to load/unload all my scenes.
This worked for that project because streaming in multiple scenes worked for that project; I was bringing in a lot of smaller elements to make as ingle larger "world" in my game.

But I don't want to do that for my current project; given the scope of my levels for this project it would be too much of a hassle to set up multiple scenes in the editor. Unlike my last project I'm only working with one scene at a time, so I don't want to have to track multiple open scenes in the editor. I want a way to consistently get the same UI for each scene individually.

I suppose I could try making the entire canvas a prefab, but something about that is setting off warning bells in my mind. I don't actually see something wrong with doing that, but it sounds wrong.

Also, how can I automatically set up all the connections? The HUD needs to know the values it is displaying; I'm wondering if there are any good ways to get my scripts to be able to find those elements without me having to plug them all manually in the editor.