When you shoot a rocket into low earth orbit, I know it’s most fuel efficient to start the gravity turn relatively low in the atmosphere, usually well before max-Q is reached.
When your target orbit is something way beyond LEO, is there ever a situation in which it’s more fuel-efficient to start your turn much higher up? Perhaps even not turn until you’re past the Karman line?
No, because you need to raise both sides of the orbit, and the closer you are to Earth’s center when you burn your engines, the more efficient they are.
The answer to your question is no, and for the same reasons as you make start to make a ‘gravity turn’ early in trajectory; that is to say that what is important in achieving a specific orbit isn’t how quickly you go up but how fast you achieve the required orbital momentum. Essentially, all of the impulse that is expended while you are trying to achieve orbital speeds is just wasted propellant and energy in the sense that all it is doing is holding the vehicle up from gravity, which is sometimes referred to as “gravity drag” because it is a direct loss of performance analogous to aerodynamic drag. Once a vehicle is at orbital velocity, it is no longer necessary to ‘waste’ any impulse in holding it up and you can shape the trajectory to optimize propulsive energy.
As a practical manner, most launches to higher orbits are not done in one singular maneuver. On a typical two stage launch vehicle the first stage will provide most of the impulse to achieve orbit and then be expended, and the second stage will be used to push the payload into the desired initial orbit. What happens then depends largely on the specifics of the payload and mission design. For some payloads, they deploy and power up immediately for checkup; others will begin a “barbecue roll” to make sure the payload is evenly thermally conditioned. Some payloads, like spysats, won’t fully wake up but will telemetry back some basic health & status (H&S) data to indicate that the payload is intact and ready to deploy. It is convenient to do all of this in LEO where coverage is guaranteed (especially if using TDRSS) and communication latency times are low; once a satellite is in higher orbit there may be gaps or unacceptable reductions in link margin to make get telemetry and make corrections.
Almost all higher orbit payloads will have some kind of coasting phase where the initial orbit is evaluated and fine tuning of higher orbit insertion parameters are refined. After that, the upper stage either ignites or a ‘kick stage’ will push the motor to the higher orbit in an orbit-raising maneuver, and either refine the orbit as it goes into insertion or the payload will do that itself if it has onboard propulsion (as virtually all MEO and HEO satellites do). Some interplanetary payloads go out to interplanetary injection in one maneuver because they need a specific alignment to achieve C3>0 and there is no value in loitering in lower orbit, and I’m sure that if you look around enough you can find some satellite that went directly from ground to MEO in a virtually continuous maneuver (probably something like a Molniya or significant retrograde where you’d want to avoid crossing more standard orbits) but anything with a near-circular final orbit is almost by definition going to end up performing at least one circularization maneuver, and maybe a series of inclination changes depending on the original launch azimuth, although those are very expensive maneuvers and it is much preferred to launch as close to your intended inclination as physically possible.
Missed prior to posting, but while the Oberth Effect certainly comes into play when timing orbit raising maneuvers for maximum use of impulse, the reason for doing a gravity turn as low as possible is per the explanation above. The o.p. mentions that the turn is typically done before reaching the “max-Q [alpha]” condition (the point of maximum dynamic pressure in flight) but that also not strictly relevant to when the turn is doing; max-Q alpha is primarily a concern for the dynamic loads (aeroacoustic vibration) and forcing bending modes on a long slender launch vehicle that can result in controllability problems but you really have the same issue whether you are going straight up or sideways.
Stranger
Just curious, @Ponderoid , but what are you trying to reach that’s way beyond LEO?
(I ask because just today I went through a bunch of delta-vee and patched conics calcs in support of a university project. with the figures fresh in my mind, I’m especially curious as to what you’re trying to do.)
Thanks, Stranger. A very thorough answer, and this part was very helpful in understanding it.
I was blue-skying some plot ideas for a sci-fi story in which someone wants to launch a rocket, but they need to avoid flying it into airspaces where it isn’t allowed. For other plot reasons, they’re very constrained in potential launch locations. Practically, there is an upper limit to a country’s controlled airspace. What if those rockets were to just fly up high enough so they were outside traditional airspace limits when they turned? Perhaps there might be an economic niche for satellites in higher orbits. Big if, I know.
Curiously, not only is it more efficient to burn as deep in the gravity well as possible, but under some conditions it’s more efficient to raise your orbit past the target orbit, then raise your periapsis (low point in the orbit) when you reach apoapsis (high point), then finally lower the apoapsis when you get back to the periapsis.
This is called a bi-elliptic transfer, in contrast to the more usual Hohmann transfer (which only takes two burns), and is more efficient when the ratio of orbital radii is more than about 12x.
I think the OP is studying up for Kerbal Space Program 2 (just a few days away!)…
I haven’t finished with KSP1 yet, but I’ll definitely be watching for when KSP2 goes on sale. 
In that scenario the obvious solutions (that probably don’t work in the premise of your story) are either sea launch or air launch, where you can place your launch site out in international ocean in a trajectory that doesn’t overfly an unfriendly nation. Those are incredibly difficult problems with their own limitations but still easier than trying to make a launch vehicle fly straight up and then lean over.
Stranger
As I recall, once you are above the atmosphere, infringing on the airspace of others does not apply.
Transfer from LEO to higher orbit requires at least 2 burns. All orbits, when you fire the engine, return you to the same point that the engine turned off. So essentially the most efficient energy wise is a Hohmann transfer. You achieve an orbit as low as possible, burn tangent to the orbit to raise the far point (apogee) then when you reach there, fir tangent again to make it circular.
…Unless you have a low power long burn motor, where you perform a spiral climb outward with constant tangent thrust - or you do it in stages, which is simple spiral using steps. Essentially, a circular orbit is “travelling at a constant X m/s” where that speed balances gravity. To achieve that, you need to expend the necessary net energy, E=(1/2)mv^2 If you have power to waste and/or time is of the essence, you can over-expend and correct by slowing down.
So the question is, how high is “safe” in your scenario to start going into foreign airspace? Ignoring atmospheric drag, the lower you start building horizontal velocity (I.e. tangent to gravity) the less energy expended just going up (the vector perpendicular to gravity) the better - as Stranger points out, it is just pushing you up against the draw downward, not contributing to achieving orbit -turn off the engine and you return to whence you came, accompanied by Rapid Unplanned Disassembly. The tangent vector is putting you into an orbit - when you achieve a sufficient horizontal velocity, you don’t fall fast enough to hit the earth, you fall “around” it as it curves.
I wonder how long it’ll be before they have MechJeb 2 out?
I hope they incorporate something like MechJeb into KSP2 at launch.
I know, right?
MechJeb was already MechJeb2 in KSP1, I think someone gave up the mod and it was picked up by someone else early on.
Regardless, KSP1 was getting modded all through pre-release, and I hope KSP2 does the same thing. I’m not sure about autopilot, but at least the deltaV readings will be included now.
If the OP is trying to avoid overflying someone else’s territory during launch–which is a valid concern–they might want to look into a few things.
Edited to add: Once you are in orbit, then you’re pretty much free to fly anywhere, as is customary, provided you avoid hitting any other orbiting objects. But there are no “no fly zones” in orbit, in terms of position over the Earth. Well…except you may want to avoid flying continuously over 800 km in altitude because of the radiation, but that’s another thing.
That probably depends a lot on the rules for different airspaces, and whether they have the capacity to shoot the rocket down or cause the launcher legal trouble at whatever altitude it is when it leaves the airspace it did have permission to fly in. Or would they bother to squawk at all once they see it’s way above the ceilings of any aircraft and it’s not pausing to spy on them.
Those details will probably be critical to the eventual plot of the story.