The original question was about fuel flow reduction with constant "power" but a change in altitude. That's really just a theoretical question, good for understanding but not as applicable as what is being discussed now.
All the performance chart information, wind calculations, practical experience, and rumours just can't replace a fuel flow gauge and groundspeed. Fuel flow divided by groundspeed tells you how much fuel you use to travel 1nm. Find the speed that minimizes fuel/nm and you've got the best range speed, for the conditions you're flying in (i.e. wind, altitude, bleeds, etc.).
Of course, climb schedules won't jump out of that calculation, but I'd recommend the AFM for your climb schedule. In fact AFM will probably be validated if you use the trial and error method of minimizing fuel/nm.
If you don't have the effect of wind in your max range performance charts, then speeding up in a headwind and slowing down with a tailwind will improve your range. The question is how much? Rule of thumb is 1/2 the head/tail wind component, but you'd need power required curves to properly calculate it on the ground.
I'll probably generate some discussion with this point, but for most helicopters, the max range speed occurs around the speed where tip effects start to influence your power required curve. (Tip effects are where the speed of the advancing tip is fast enough that compressibility of air becomes a concern). Of course airframe limitations may not allow you to reach that speed, in which case Vne would be your max range speed, or whichever speed max continuous power gives you.
Matthew.