Without knowing what soap you used, I'd recommend using Dawn (the blue stuff) and a soft scrub brush. That will get rid of any remaining oil.

You might also make a paste of oxyclean and Dawn to spread over the stained areas. Be sure you clean the back and the front sides. Let it sit for a hour or so.

You could also try hydrogen peroxide to remove the stains. I would soak several old cleaning rags with the peroxide and just let them sit on the stains for awhile.

All of that said, getting stains out of plastic is difficult. Again, just make sure you're cleaning the front and back sides and rinse with a garden hose sprayer so that the water really goes through both sides.

you might also try soft scrub with bleach or something similar. Or liquid barkeepers friend

First of all that's not rattan....that's mesh.
I second the Dawn....or just spray it with something like 409...brush it with a soft brush and hit it with a stiff spray from the hose.

Yup, Phifertex. How to clean: How to clean Phifertex