I seem to recall from the dim and distant past when I did the CFS course in the RAF that the requirement for maximum range in a jet was
1/SFC x IAS/Drag x TAS/IAS
1/SFC (SFC being Specific Fuel Consumption) is an engine requirement and is met by operating the engine at its design RPM (or equivalent) which is usually around 93%. Ideally therefore the aircraft should be climbed to an altitude where this RPM gives a cruise speed that satisfies the other two requirements. As weight is reduced the aircraft is allowed to cruise climb while maintaining design RPM.
IAS/Drag gives the best ratio of speed to drag and is derived by drawing a line from the origin to a tangent on the total drag curve. It is usually about 1.3 times min drag speed.
TAS/IAS is the reason that jets are more efficient at altitude. Flying the best IAS/Drag speed at higher levels increases the TAS/IAS ratio.
So, to get the best range immediately after take off (in an ideal world) the aircraft should be climbed to an altitude where design RPM gives the speed for best IAS/Drag ratio, the altiude thereby also giving the best TAS/IAS ratio. As fuel is used and weight decreases, min drag speed also decreases and with it the speed for best IAS/Drag ratio. The secret then is to allow the aircraft to gently climb at constant TAS while maintaining design RPM. This has the following benefits-
Design RPM and therefore engine efficiency are maintained at the optimum.
The IAS/Drag ratio is maintained at its optimum, although the IAS will gradually reduce.
As the IAS reduces the TAS/IAS ratio improves.
All of the above assumes no penalty from compressibility, but I did say that it was in an ideal world.