Sign with broken base

I was called to change a sign that was broken at the base. Original sign was a round pole and about a 3" core drilled hole. I ran a 3.5" core over the old hole and pole. I ran the drill all the way down (about 12") and could not get the pole out. What is the proper way to replace a pole broke at the base. I ended up core another hole right next to the old one. also when you guys core a hole for a new sign, what is the fastest way to dig the hole. We have always used a breaker bar but the dirt has to be removed by hand.

Whenever we have a post broken at the base we put a flat bit on a jack hammer and put the edge of the bit on the edge of the broken post and fold it inside of itself below grade. Then patch with cold mix AC or cement/mortar. You can mix lamp black with the cement if you want it to match the existing AC. Trying to core drill over a broken post you will more than likely be trying to core through 2 feet of cement also. Aint' gonna happen. We use a core drill for installing new posts. When installing round pipe we use a post hole digger that has a tee handle and you twist. Works real good removing dirt after the core is removed. Also, if one is available, a gas powered auger works best. Just use an auger bit slightly smaller than core drill bit. Don't forget to notify your local Underground Service Alert. Not too many people realize this, but, it's the law even if hand digging. Fiber optic lines can cost hundreds of thousands of dollars to repair.