I'm familiar with that type of fence material, so, depending on a few things, here's what I would do (despite the fact that I would NEVER set posts in concrete because of this replacement problem). The post spacing is important because of visible appearance and also where the splices are, so it may not be practical or possible to dig an offset hole. So:
Remove the old concrete slug. Dig around it enoung to wrap a chain around it and try your tractor's 3 point lift ability or even drawbar force but be careful, this could flip the tractor or hurt you if the chain breaks. Or, chances are the cement was a weak grade used for pouring in dry powder and letting it setup using ground moisture or a bucket of water. If so, drive a steel bar into the post hole and crack the plug into sections which will then be easier to remove. You can sometimes get good results by pouring muriatic acid into the post hole to dissolve the concrete or weaken it enough to break it apart. if you get all the wood out, chances are that the new post wil swell and crack the concrete anyways. A couple of .357 or .44 mag rounds installed at an angle will also contribute to the concrete's demise. If AP rifle rounds are legal where you live, they work really well, too.
You didn't say how deep the concrete plug goes down or what diameter it is, but just splitting it in half (as seem from the top) and removing it would be my simplest 'fix' suggestion. Getting a new post into the old hole is not likely to be worthwhile, because it would need to be shaved or driven (pounded) into position. I'm surprised that you have not found a neighbor, friend, boarder or renter to ask for help. Two (or more) hearts and minds are better than one in these situations. Try to come up with the easiest solution because there's going to be more posts broken before you retire. This includes whatever new method you need to set the post (concrete or not).
