upholstry

The factory had a press over the work table. This allowed them to compress the filling and springs, so to attach the skin tightly. When turned loose it filled out properly.
You will likely find the filling in you seat has depleted with age/use.
Result is often a loose wrinkled skin. If I had a nickel for every car I've seen on for sale with "new Legendary seat covers" in the description and poor install in the pics...
Anyway... How I do it. First I refurbish or replace the filler.
Next I place a piece of 2X8 across the back of the assembly so a couple of helpers can stand on it, mimicking the factory press. Skin goes to the attach points easily.
A clean work surface and area is the most important part. If you should put a new skin down on a remnant of a hog ring.... cursing can result.
I take the seats apart and prep for reassembly outdoors. I put them back together on carpet in the living room floor whether wife likes it or not.
One more thing... beware those threaded anchor studs on the seat tracks. The factory had rubber covers over those until inside the car, removed at very last minute. I have found a few of them left in the car. A bare stud can scar you, your door panel, anything it gets near. My UPS delivery guy got his leg ripped open by one sticking out of a box containing a bucket seat. His going to get a tetanus shot after work on Friday equaled GRRRRR.