How to chase helicoiled threads?

Cleaning up some stock heads off a 360 and chasing all the threads. Both of the end holes that go into the water jacket had been helicoiled but so dirty I didn't even notice until I got the 5/16 chasing tap into one of them (got a few turns in, felt pressure, stopped, and tried to remove). The helicoil started to unravel so I pulled it out. Is there a way to chase the other hole, which is filthy, without destroying the helicoil?