You can get sticky back felt. I'm in the UK, you should be able to get it in the US also, on Amazon. I stick the piece to the felt first, then cut round the piece. Works perfectly.
Thank you. I think I'm getting old, because it did not even cross my mind that there might be such a thing as sticky back felt.
Yep, I reckon if you find excellent billiard table Baize, some wood glue and sharp scissors - you’ll love the outcome. I think new felt is a wonderful upgrade. It’s so good that I am doing everything in my power to talk myself out of re-felting all my old sets - even a 90-100 year old French Regence set … it’s probably sacrilege… but gee it would look and play better and it’s not like a museum piece worth heaps … but still, I probably won’t as it’s the closet thing I have to a proper antique chess set and I want to respect its history.