This must be a common problem and I'm surprised there weren't more replies.
My Silver Pigeon 1 came with a grippy recoil pad which I found would not slide over my clothing, my old Winchester 101 XTR had a hard plastic plate so the difference was noticeable and distracting. At the time I found myself adopting a style where I held the gun away from my body slightly and then pulled it directly into my shoulder to stop it gripping my clothing, which I agree is not ideal.
These days I don't think about it and, to be honest, I'm not sure if I still hold the gun away while mounting or if wear and natural patina has taken care of the problem, I suspect the latter. If I think about it over the weekend I'll take a look and post an update
p.s. I did take a look at my gun and the rubber recoil pad is still a bit sticky so, as I'm not having any problems with it, I can only surmise that I have subconsciously modified my style to suit.