This is how I do it: -
-With the tapered brass pin removed, reassemble the stud into the balance cock and secure gently with the screw.
- Install the balance wheel complete with the hairspring onto the jewel in the base plate.
- lower the cock into position, lining up the pivots and install the screws
The balance will now be free to rotate un-tethered by the hair spring, but try to avoid this, as damage can occur to the free end of the hairspring. Now, carefully rotate the balance wheel so that the free end of the hairspring travels towards its home-to-be in the stud hole. If all is lined up correctly, the free end of the hair will now go straight into the stud hole, like threading a needle. Do not try to move the free end of the hairspring with tweezers if its off the target. Its hard work, and ends in tears. Instead, rotate the balance backwards, thus moving the free end of the spring away from the stud and creating yourself some working space. Gently tweak with tweezers. rotate the balance wheel forwards again and repeat process until that hairspring threads itself through the stud with your eyes shut.
Now, with the spring engaged so that its just starting to show on the other side of the stud hole, you can now reinsert the brass taper pin and lock the hairspring to the stud.
Note that due to the hairspring now being slightly shorter than before, and all other things remaining the same, in theory the oscillation (vibration) rate will be faster so you will need to regulate the watch again. Also, the roller jewel will be slightly off center position (watch out of beat). Does your watch have beat adjustment? If not and you are really trying to perfect the movement, then you need to tweak the hairspring hub on the balance staff. Mark Lovic beautifully describes this in a video on here - but I cannot find the link just now. Can someone assist please?
I hope this helps?