Those plastic units that roll between a washer and dryer have pull out drawers and a flip up top. You could put sewing supplies in the flip up section and pieces in the drawers. The drawers are three different sizes. Don't know the height of your table...but they could roll under it maybe.
Looking at your first post, what about a narrow book case between your table and your desk.
I also have those big units that WoodlandWoman mentioned.
I sort my fabrics by subject/style: baby prints, Civil War, 30s/40s era, then combined collections by color both solids and prints (Creams/whites, pink/red, orange/golds, yellow, green, blue, violet, brown, black).
I also have a container just for my large cuts..5 yards or bigger that would be suitable for backing.
A container for flannels. I also have a container for batting end pieces. I purchase my batting when Handcock's has their 40% off sale and get a whole roll...I get a senior discount also so that helps. There are always end pieces that can be used...sometimes for small projects and sometimes, I can join them for larger projects.
Sewing tip: lay lengths of batting on a table, covered by your cutter surface. If the edges don't match, overlap slightly and use your roller cutter to make a clean cut. Butt the two sides together and hand baste the two edges with a zig zag stitch in a neutral color, leaving a smooth surface. I fold in the sides and then roll the piece to store it so not to pull the basting. It's surprisingly sturdy enough to hold up well for both hand quilting and machine quilting.