Air-augmented rockets have not historically been very successful. In most cases, adding a shroud around the outside of an existing rocket was a large weight cost in exchange for only a modest increase in thrust specific fuel consumption. Because they weren't optimized for using the air as reaction mass, most of the added thrust was the result of secondary combustion between the fuel-rich rocket exhaust and the atmospheric air, making them essentially very inefficient ramjets.
If, however, an air-augmented rocket engine were designed in an inside-out configuration with a central bypass rather than an external bypass, you'd could end up with a simpler, lower-drag design which could be scaled up and would likely be pretty high in efficiency.
Such a design could allow a really, really high bypass ratio, causing thrust specific fuel consumption to drop ridiculously low. The combination of really high thrust and really high specific impulse is pretty nice. How high of a bypass ratio would be possible before diminishing returns made the added drag too high?