I know the clips that you speak of. I think that the "clips" are actually wires themselves. They keep the horsehair pad attached to the main wire on the frame.

When I took this one apart, I had no clue how to remove those "clips". Seems lke a special tool is required because they have to pass completely through the horsehair pad and grab the main wire, with some tension.

This wire runs right along the inside of the bolsters. So, I added some foam inside the bolsters. However, this is not the best solution because it now renders the bolsters ineffective. If I do it again, I'd like to properly replace the horsehair pad. But, which tool to use to remove those "clips".