The normal way to restart an engine in-flight is to use air from the APU to turn the N2 section to sufficient speed to start the engine (15% N2 on the CFM56). Since the restart is only dependant upon the core speed, you could argue that high bypass engines are easier to (non-windmill) start than pure jets because the core engine is smaller.
If you can’t start your APU, and most APU’s can be started near the ceiling of an aircraft, then you will have to windmill the core to restart.
The parameters of a windmilling engine are a function of speed. The two conditions I see regularly are FL150, 150kts which gives about N1=10% and N2=0% and FL240, 260kts which gives N1=18% and N2=15%, although these would be much lower if allowed to decay down to the steady state. It takes many minutes for this to happen even when stationary as you can see when shutting down on stand. So at an IAS of 150 or 260kts the wind-down is much slower, this means that you will probably not have to windmill from zero. Furthermore, there comes a point when the engine does not wind-down because the IAS is so great and for 15% N2 (the minimum restart N2) that is about 290kts.
So back to your windmilling restart, if your engine runs down and you are happy to attempt a restart but have no APU, you will probably still have enough N2 to attempt the relight even after you have gone through all the checks. If you happened to be working slowly (not a bad thing) or were at low IAS when the failure occured, then your N2 may be below the minimum for relight. In this case just lower the nose to increase the speed to above 290kts and when the N2 reaches 15% raise the start lever and away you go. With MCT set on the other engine this wont use up much altitude unless you started from zero N2.
Finally, you ask about new devices to improve in-flight restart. The 737-NG’s have an auto-relight function which will automatically restart for you under certain conditions (flameout not failure). This allows the relight to happen before the N2 has decayed below the minimum restart value thereby eliminating the need for windmilling dives.
S&L