This is SO easy! I hope you can get some mileage out it!
Here is the small tail. It is too small to thread on a needle and then be able to get the needle into the hole the thread makes.
Take a contrasting piece of thread and thread it into a loop on a needle.
Insert the needle in the hole where the small tail sticks out of the fabric.
Put the small tail thru the loop.
Tug on the loop. The loop will bury the thread. Just pull the loop all way out. It has done its work! Trim the small thread. VOILA! All done...