I used to work as a rigger splicing rope 40 hrs a week. For the small diameter line what i would do is instead of shoving a fid down into the core i would use a piece of coat hanger bent in half to make an eye. I then go up the core from the bottom of my bury and tape the already tapered tail to the eye and pull it through. On the small stuff it is impossible to get a fid through far enough to do a proper bury.

hope that helps