You are trying to use the pin to hold the string. The pin is only there to lock the string into place.
The ball end of the string should lock up under the bridge pad. The pad is a piece of hardwood that is directly under the bridge. It's purpose is to minimize effects of bridge rotation. Rotation causes a hump behind the bridge and a belly in front. The ball end should come to rest on the pad rather than at the bottom of the pin. When a guitar has a round sound hole it is easy to reach inside and feel the pin and the ball end and manually seat the ball against the pad. If you have thin arms you may be able to reach inside of a Thunderchief and do the same. If the ball is sitting at the bottom of the pin move it and pull up on the string at the same time. If you cannot reach in then try inserting the the pin backward a small amount while pulling up on the string. When the ball end comes to rest against the pad spin the pin around and insert to depth while keeping upward pressure on the string. When properly strung the guitar will have better tone and dynamics.
BTW, if you've ever seen a bridge that is split along the grain through the holes the reason is repeated forcing of the pins into the holes during restringing. Makes sense, when you think about it. A bridge pin is just a conical wedge. It is no different than splitting logs for the fireplace.
thuncerchief string changing.. 
