To a certain extent. This would allow more RPM hence
more HP to be applied
to the prop in it's lower efficiency range around
take off speeds.
The prop efficiency curve would still apply and that
is sort of the
same shape as a quarter sine wave. A variable pitch
prop, on the
other hand, moves it efficiency curve downwards
towards lower speeds
with a reduction in pitch. It is quite possible to
have max prop
efficiency (sans effects from non optimum twist)
with a variable
pitch prop at lift off speeds and every other speed
up to top speed.
The ideal variable prop would change both pitch and
twist.
So it is better to forget shifting gears and switch
pitch instead :-)

Is this clear?

Paul Lamar

Strange as it might seem, the variable pitch prop
really reduces prop efficiency at low speeds!! What
the prop really does is allow you to achieve more
horsepower from your engine at this point - which
increases disc loading for a reduction in efficiency.
The increase in delivered power more than offsets this
loss in prop efficiency (limitation to this statement
in that it assumes that the prop was not hoplessly
stalled before you reduced the pitch, and actually the
times that a propeller is stalled is surprisingly
few). This same case would apply if you got more power
out of your engine by "downshifting". With the stupid
way prop efficiency is computed - the easiest way to
improve prop efficiency is to throttle back the engine
- does not make a lot of sense since that will slow
the plane down. Ah the magic of modern aerodynamics - Vance

This chart is from Martin Hollmann book Prop and Duct Design.
Advance ratio is another way of saying pitch sort of.
You can read the x axis as aircraft speed sort of.
Notice as the pitch is reduced the max efficiency speed is lowered.
Notice also if a fixed pitch prop is chosen for cruise the efficiency 
is very low at low speeds.

Paul Lamar
 
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