What you are doing at the moment was what caused me years of unsatisfactory upholstery results. Finally, I needed to redo my Camelback sofa with rolled arms and I asked my Mom to come help.
It took literally less than 10 seconds for me to see what I had been doing wrong for years.
I kept trying to "stetch" the fabric to fit the item - much like stretching a canvas for a painting. That is not how it is done.
The fabric must first be "trained and coaxed" onto the item. Once the fabric is trained into place - with no puckers - only then do you tighten (pull) into place and staple or tack.
From the looks of the chair in the photo, it seems that the back was not originally upholstered, as there is no padding covering the wood. You always want some padding between the wood and the upholstery fabric to prevent the fabric from wearing too quickly.
The back first needs to have a backing to hold the padding in place. This is typically done with jute webbing that typically comes in strips approximately 3-1/2" wide. These strips are stapled into place extremely taut. There is a special tool made for this, but I have done this without the tool, or you can make one with a chunk of 1x4 and a few screws. Then a layer of foam or wadding is placed on the back. 1" foam would probably be good for this. Then, over the foam is a layer of polyester batting, made specifically for upholstering. For the rear back, I would probably first staple a layer of non-stretch heavy poplin or denim into place and add a layer or two of quilt batting.
After all of the padding is in place, then the fabric is placed on the back and you begin to "train" the fabric by repeatedly smoothing the fabric with your hands. You would start center back and work to the outer edges by radiating the movements equally around the back. I hope this makes sense.
To understand the movement, try to smooth a pair of trousers over your leg using the palms of your hands.
Now that I've tried to explain this verbally, it occurs to me there is probably a YouTube video that would better demonstrate
I'll take a look.