One thing to throw into the discussion is that when speaking of efficiency, we are talking about a high bypass ratio jet engine vs low bypass. All modern jet engines with high efficiencies are the high bypass ratio. Look at a B777 and the diameter of the engine nacelle is huge compared to the actual engine. I can't remember the exact ratio but I think it's something around 80% of the thrust is generated from the air going around the engine, not through it. A rocket would be like sitting in a boat filled with tennis balls, a gas powered tennis ball serving machine and the fuel to power it. So the boat has 1 ton of tennis balls, the gas and the machine. The faster the balls are served out the back of the boat, the greater the amount of thrust you generate. Or you connect the gas machine to a propeller, remove the weight of the tennis balls and expend less energy using the medium around you (the water) with a much lighter boat without the weight of propellent, the tennis balls. If you operate the propeller at the same fuel consumption rate as the tennis ball serving machine, the propeller would move the boat faster than throwing out the tennis balls. Especially initially with the weight of the balls in the boat. If you could compare the very last ball thrown vs the amount of thrust generated by the propellor during that same period of time... I'd say it is possible maybe even likely that the throw of the tennis ball on the last one generates more thrust per unit of time than the propellor, but during the majority of the test, the weight of the balls would cancel out that perfect conversion of potential energy to thrust. (just a gut feeling on that, maybe a seasoned physicist could chime in on that). I think the drag generated by the propellor moving through the water, the rotational drag of the propellor, turbulence around the propellor and the thrust vector being off axis would all contribute to inefficiencies that would not be present in converting mechanical power from gas engine to thrust. Throwing the tennis ball is like a rocket with more direct conversion to thrust.
No wonder spaceship one uses a Jet to get things started. A rocket at rest that must go vertical has an enormous task of lifting itself from a velocity of zero when it has 100% of the propellent at start. Or you take it to altitude with a jet, drop the rocket which is already traveling at say 400mph, simply modify the direction vector with the elevator, then ignite the rocket. You didn't need any of that heavy propellent to get you to altitude and speed. Think about the space shuttle. The majority of the fuel is used for the first 25% of the flight. A little off track, but I think looking at those two examples shows the strengths of each. Inside the atmosphere the Jet is more efficient, in space... the rocket wins. :)