You can buy taps with different lead tapers on them. You probably have a taper or intermediate tap. You can buy what is called a bottom tap which has almost no taper on the lead in.
Get all the holes started with the intermediate tap you have and then Buy yourself a bottom tap to finish the holes off. A better option than destroying a good intermediate tap.
edit: I found this picture which shows the different lead tapes on the taps:
Cheers