It happened once, it will happen again.
I'd start by getting as much of the old post out as possible. Perhaps a hole saw on the end of a longer shank would chew out the center. Another option is a larger forstener bit to munch the remains of the post.
(Updated - not recommended) If you're competent with a chainsaw it may be possible to plunge-cut downward and eat out the post from the inside. This needs a long bar and a chain you're prepared to damage if it kisses the concrete. Plus kickback could be horrendous. (I'm no chainsaw user, check with someone qualified)
A shop-vac would help to pull out dislodged pieces and keep drilling.
A crowbar or pry bar would help push splinters off the walls and corners of the hole.
You might be tempted to try and burn the wood in the middle out - however that runs the risk of cracking the concrete, presuming you can ignite a fire with limited air supply. Plus it will be treated wood and that's both hard to light and releases nasty chemicals once it is burning.
When the hole is clear, use a spokeshave to subtly taper the new post for the depth of the hole, and then sharpen the bottom 2 inches/50mm to a 90 degree point. Drop some sand or gravel, or some wet concrete into the hole and fit the post. The gravel will form a "cup" to reduce rocking due to the slight taper. Concrete will do the same but will form a socket.
Next time the post needs replacing, simply bring up your old measurements and carve a new post with a taper and point. Then either just lift out the stub by hand, or screw an eyebolt into the stump and lever it up and out.
You may need to cut one or two thin wedges to hammer into the post hole beside the post, to take up any slop. Or you could use a sealant around the top to try and minimise rain/snow ingress.