"so now, maybe you are on to something that will work for you. Are you going to be able to hook this tray onto your frame?"

I do think that it will hook onto the frame or the roller bars - if not Amazon has an easy enough return policy - although this basket does seem really useful! (not too expensive either - $20)

Remember those boxes that checks used to come in? Wonderful!! It's small enough to be portable and actually can hold "a lot" of stuff. It's easy to put on top of the quilt as you move along or to put next to your chair on a table or the chair. The box I am using holds a magnetic pin/needle holder, thread, thread clippers, a 6 inch ruler, mechanical pencil, thimble, a pkg of extra needles. Love it. All of the essentials are together in one place and it's not in the way--if it is, it's easy to move, yet accessible. Other than a template, what else do you need besides needle, thread and a thimble?
Now that I am rereading your post, you comment that your frame sits at an angle. That's probably the problem with having something such as any container placed directly on the quilt. Depending on how taut you have the quilt, even my suggestion might not work. Yes, something to sit beside you will be the best. I'd want an open container so I could easily see what I'm needing for the next step such as rethreading the needle.

How about looking in the cleaning section of a local box store? I bought one of those containers with a handle that have several sections in them. I used it when I had a cleaner coming in. In each of the sections I placed the type of cleaner in it for a specific area. I had things like scrubbing bubbles, windex, floor cleaner, toilet bowl cleaner in it. you would have options as where to set it, on the floor, by your chair, on the table, underneath the frame. Of course you will have to bend over to grab the handle. I also remember my grandmother had a little folding cloth bag that had a wooden frame and opened like a TV stand. She kept all her knitting supplies in it including the project. It was easy to carry around and place it where ever she went. Oh, and for my cleaner I also put the sheet of paper covered with a plastic cover rolled up in one of the spots and put my list of what I wanted done and what cleaner to use.

I stored my stuff in a tin box on top of the quilt and used a big magnet underneath to hold it in place.

Neat idea, I was going to suggest T-pins attached to a small card-stock box, just stab it into the quilt where needed.

Ohh Polkweed
...you are officially brilliant... My husband says my engineering degree is wasted on me....lol...that sounds like just what I should do! Thank you! Nina

Yes! I use a vintage Whitman candy tin with strong two piece magnet to hold it to the quilt top. I love the ease of magnets, I use them for everything possible.

Forgot to say the tin has the hinged lid so it stays closed when I'm not using it.

I recommend getting a magnet with a built-in hook so you can use it as a handle.