Here's the best explanation of the process I've ever read. It's written by WV (David), a learned HF member and DIY guru. This was the "ah-ha" moment for me. Once I got it using WV's instructions, I've really enjoyed making whoopie slings and have never had any trouble. NEVER NEVER NEVER use tape on your tapers to try to make a fid out of the amsteel. It gums up and gets stuck and makes a mess (in my experience).
"If I understand you correctly, you may be using too short a taper. I pull out two strands about 2" from the end and cut them off as close to the main rope as possible. Then I pull two more strands about 1.5" from the end and cut them off. Then I poke the splicing tool of choice backwards through the eventual bury and grab the tapered end about .75" from the end. That way the small ends of the taper fold back, but don't reach the point where the first strands were cut off, and the doubled portion you pull through is the same diameter or smaller than the main rope. It also helps to enlarge the hole where the splicing tool exits the bury by pushing the rope together to loosen the braid and then pulling it against the splicing tool in both directions to make the exit hole larger. (Do this before you grab the tapered end.) When you pull it through, grab the bury and milk it back over the tapered end.
This is the fail-safe method. In practice I often use a shorter taper, but do it this way first. Once you get the hang of it you'll adjust to suit yourself. Good luck!
I use floral wire and I recently made a tool that works great. I doubled up a 3-feet long piece of floral wire and wrapped the ends around a short piece of tubing.
The tool is shaped like this: ------------------|
The working end of the tool is the loop of floral wire with a soft "nose" for spreading the strands. The tube keeps the amsteel from sliding off the back of the floral wire.
Good luck. It's fun.