In going along with what Chuck said, along with buying a split die, one that you can adjust the depth of the cut, I would also buy a chasing die. These aren't split and are hex shaped so that you can use a wrench on them. They also have to be turned around to get close to the head. You have to make sure before using this die that the threads are full depth in the piece you are fitting it to.
The trouble with recutting small screws close up to the head is that the holder of the die is usually in the way of getting close to the head. One way to do it would be to use the holder to start the thread, then take the holder off and use a pair of pump pleirs on the die and chase it to the head. The die is hard enough that the pliers won't hurt the die. Just don't put a lot of pressure on the die with the pliers.
You would need an adjustable die, to keep trying the fit to get the right depth in the thread
Another way would be to buy an acorn die and holder. The holder is adjustable in the deph of cut, not the die. In my opinion the acorn die is easier to use.
