Changing Connected Clip Anchor Points?

A connected clip has a small line "anchoring" it to the storyline. Normally this line is at the start of a connected clip. However, either by design or due to another Apple bug, sometimes this line occurs within the connected clip. This can happen when trimming a connected clip to extend its starting frame after it's been placed. The "connection point" remains where it is relative to the storyline instead of moving with the connected clip's first frame. I can see where this would be desirable at times, but other times the connected clip's new first frame is over a different clip in the storyline than the original connection point and the connected clip meant to be relative to its start frame rather than its original placement point. (Reducing a connected clip's length by trimming the front smaller instead of larger moves the connection point to stay with the connected clip's first frame.) Is there a way to Trim a connected clip larger such that the connection point moves with the clip's start, or move the connected clip's reference point? I tried clicking and dragging on a clip's connection line, but that apparently isn't supported.