I've always found the best way to tackle banjo head tensioners is to go round them all in turn with an appropriate head key and check that none are either loose, or less tight than the others.

Once you're sure they are all at the same tension, go round either clockwise or anticlockwise giving each one an eighth of a turn until you're back at the place you started - as suggested above, the neck is as good a starting place as any.

If that isn't enough tension, repeat until you're happy.

The bridge will always deflect the head to some extent, so don't try to correct that.

If the head is real skin, take care as it may split, but a plastic head is virtually indestructible!

A plastic head should not split away from a metal hoop... but don't attempt to get much tension out of a cheap head with a plastic hoop - they are bad news!