I have always taken the new string once fed through the eyelet of the tuning machine and put a crease to the string tail using the backside of the eyelet. After that, I wrap the tail under the string at the front side of the eyelet and form another crease before winding the string around the peg. This locks the end of the string inside the winding and drastically decreases the chance of slippage. This method is common, but I've seen many that don't use it.
It really helps to keep me in tune.