Practical and useful - yes. Economically viable or technologically possible? Not sure.
Surely it wouldn't require nearly as much materials so incredibly durable as the "full elevator". Jet engine powered airplanes are vastly cheaper and easier to construct and operate than rocket-powered vehicles. Once in LEO any vehicle can engage low-thrust, low-energy propulsion like a solar sail, and depart Earth's gravity pit at fraction of the cost of corresponding jet propulsion. Essentially, that would provide the benefits of space elevator at the part of the travel where its benefits matter most - between the atmosphere (where jet engines work) and the LEO (where orbital engines work). Currently that gap is jumped with rocket thrusters that are simply extremely costly.
That much for benefits. Now for the problems.
The problems of delivering several hundreds kilometers of a space rope to the orbit aside (we've been there with classic space elevator) we have actual air drag, and no anchor that would pull it and provide energy, plus each vehicle riding up would pull it down. That thing wouldn't support itself like a classic space elevator, it would require its own propulsion to keep it afloat. (OTOH, fuel for said propulsion could be delivered by the same airplanes, and it could be any of the number of the neat "orbital engines", no need for the troublesome rockets). There would be the whole problem of docking the payload in the stratosphere at supersonic speeds. I'm not sure about meteorology of the stratosphere, but I think it could be troublesome (not worse than against classic space elevator though). And obviously the usage cost would be considerably higher than of classic elevator which could use efficient electric motors to bring payload from ground far beyond geostationary orbit - while jets are an order of magnitude cheaper than rockets, electric motors leave jets far behind in terms of energy efficiency = cost of operation.
One more problem: the active propulsion can't fail for prolonged periods of time. In case of a classic elevator, it would sit there completely inert with no problems. In case of the partial, it would fall. If lengthy repairs are needed, it could be propelled to a higher orbit, as high as needed, and repaired there over time, but unexpected faults would simply destroy it.