Yes you can overlay the nail. But....make sure there is no sign of any infection (no redness, doesn't feel warmer than other areas) Even if it stings, make sure it is perfectly clean by using your sanitiser.
It helps to 'immobilise' the split with adhesive. Put a spot over the crack and gently move the nail so the adhesive seeps into the split. Wipe off excess. Then overlay as usual.
This is a great repair as it will stop the split hurting and allow it to grow out. As usual, as your client to keep an eye on it for any changes.
Marian