If the back is relatively smooth and the surface it is going on to is pretty clean, I've had good look with transfer tape. This is made by 3m and available in most art stores. It is basically a thin layer of rubber cement-type adhesive in tape form. It is expensive, and it has its own difficulties in working with it. I used it in my job for years so I am comfortable working with it.
You can buy wide rolls of it, for maybe $100 for a 6 inch wide roll. Or I get a 1 inch roll and lay it out edge to edge. The best way to work with it is to lay the tape out on a sheet with the adhesive facing up- use masking tape at the ends to hold it in place. Then put the leatherette on the adhesive and press it down hard. Now, and this is the tricky part, take a sharp X-acto knife blade and trace along the edge. You want to score the adhesive; don't try going through the backing paper. Then lift the material up. Any little dots of tape along the edge of the material will roll away easily after application.
Licking the tape surface before putting the panel into place will make the adhesive less aggressive during positioning.
Well, I've worked with this tape for 25 years so I am very comfortable with it. And for re-applying coverings, it is excellent. Super thin, super tacky. It gets stronger as time goes on, and can be easily removed by rubbing or using another tape to lift it off. No solvents or other chemicals. You'll probably have it stick to everything at first if you haven't used it before.