With any line replacement in a mast for a small boat, I use to do either of the two.
First, I had purchased an inexpensive sewer snake and ran it thru the mast. If you do not own one, borrow one from a friend and run it thru the mast and attach your new line. Add several feet over what the manufactures says. I believe the HOW or this site store does that any way. You can attach it to the snake end with a string or fishing line. I use to take a small phillips and push thru the end and then push a small line thru and then attach. The other of course is attaching to the end of the current halyard and in some cases you will need to cut off the loop end. IF you do, tape where you are going to cut and then afterwards, burn the end so it will not come unraveled.
One trick is to have the mast elevated and wiggle it back and forth. I taught that to many which they said it helped the snake to go down easier thru the mast.
crazy dave condon