Sometimes I have that problem even when I taper the end properly.
It seems that sometimes the braid is just too tight to allow the splicing tool with the tapered end captured in it.
The solution I have found that works for me every time this happens is to take a piece of scrap Amsteel, 9" to 12" long and pull a strand out. Tie the strand into a loop, secure the loop to your splicing tool (this may require threading through your splicing tool before tying the loop). Insert the splicing tool as you would normally and just before the loop disappears into the braid, insert the tapered end into the loop and continue pulling the splicing tool through the braid.
Solves the problem for me every time.
The strand of Amsteel adds almost nothing to the diameter of the tapered end.
When I get lazy, I just leave the strand in the splicing tool and use it every time for a while. Unfortunately, the single strand of Amsteel fuzzes and becomes unmanageable and I just have to cut it and discard.