[2020] Faster way to progress through building instead of just a stone axe? Perhaps a claw hammer? Or is there more?

There is. Stone axe does 33% each hit, there's a claw hammer that does 50% each hit, and there's the nailgun that does 100% each hit. You obtain the claw hammer recipe either by finding it or by learning rank 1 of the miner 69er skill which teaches you the iron tool recipes.