Wrap the jar in plastic wrap. This keeps it clean from the glue. I wrapped it several times to even out the surface so that the cardboard tube could slide off of the jar past the ridge it had at the bottom of the jar.
Glue two strips by overlapping on the end about an inch. A single strip did not wrap entirely around the jar and this helps to glue the strips in place.
Glue a layer of cardstock around the jar. Don’t snug it up too tight but keep it close enough to conform to the jar.
Continue gluing on the cardstock strips but butting the end to the last edge glued down. This will prevent bumps in the cardboard tube. Coat the entire strip with glue to laminate it to the previous strip. Smooth over to get rid of air and glue pockets.
When you have two or three layers, slide off to test that you can get it off the jar.
Position back on the jar and continue adding layers of cardstock. I think I had about a 9 layer lamination. You get a feel for how thick your lamination should be if it is fairly stiff when it is removed from the jar. It will become stiffer as the glue dries.
When the tube is still wet, you can work the edges with something like a marker barrel or tool to burnish the edge so it has a smoother finish.