A great loop run around the perimeter of Godley Head Park.
Start at Taylor's Mistake. The trail from here which heads around the Godley Head military ruins and back to the Summit Road was rebuilt following the Christchurch earthquakes. It is a two-person wide, packed-dirt well-graded track with great views.
After reaching the Summit Road from the Lyttelton side of Godley Head just beyond the abandoned barracks the super-wide trail ends. Either backtrack or take the Crater Rim Trail (from the round-a-bout, look up the fence line to the west for a wooden style crossing the fence a few 100 meters up the hill.)
The Crater Rim Trail is singletrack, but well-graded until Breeze Col, the next point it intersects the Summit Road. It offers nicer views of Lyttelton Harbor. From Breeze Col, take the Anaconda Mountain Bike Trail
down to Taylor's Mistake. This can be muddier and several hikers/runners have opted to make a steep cut-through descent on the hill rather than follow its switchbacks near the bottom.