Frank has it right.
A haul does produce additional rod loading.
We all agree that hauling increases line speed. In order to increase line speed, the object to which the line is attached must resist the pull. This Newton's third law of motion. For every action, there is an opposite and equal reaction. This resistance of the rod tip to this extra energy added to the cast causes the rod to bend a bit more.
The haul does two things. The first is to increase line velocity and the second is to lessen mass. We commonly think of increased line velocity but also in pulling some line in, we remove some line; and this removes the amount of line mass that is aerialized. How can the double haul increase the distance of the cast if we remove line from the cast with the haul?
We increase distance because kinetic energy is Mass x Velocity squared = MxV2. Energy is what determines how far the cast will go and by increasing velocity we more than compensate for a slight loss in line mass. This extra energy allows to then shoot line which compensates for the slight amount of line removed during the haul.
It is not momentum (MV) that determines how far a cast will go, it is the kinetic energy in the cast (MVV). What momentum does is to determine the DIRECTION of the cast. Momentum has a direction component and energy does not.
What slows down the is air resistance or drag. It so happens that drag is also a squared function of velocity (V). This is why it is so hard to add that last bit of distance to a cast. The first 10 feet requires much less energy than the final 10 feet. To add that last 10 feet we must increase the energy in the cast and the velocity enough to overcome the squared velocity function of drag over the entire length of the cast. The fact that both KE and aerodynamic drag (FD) are both proportional to V2 is one of the beautiful symmetries of physics.
As to the timing of the haul, it should be during the maximum acceleration phase or maximum rod loading phase of the cast, right before the stop.
Recall that I have shown that the haul adds load to the rod, which causes the rod to bend just a bit more to resist the added energy in the line which is pulling on the rod tip. When the rod bend a bit more, the effective rod length shortens and the rod tip gets closer to the casting hand. If this rod shortening occurs during unloading, it will cause a dip or wave in the straight line path and an inefficient cast. If it it occurs too early, it will causes a dip in the rod tip path, which causes a concave rod tip path, which causes a tailing loop. So timing is critical so that the additional bend in the rod is added smoothly to the point of maximum bend to preserve the straight line rod tip path.
Consider the following articles on casting and the double haul by espert casters and scientists.
When Less Is More: Subtleties of the Double Haul Cast - New York Times
"Now you are ready for the double haul," Jaworowski said as he demonstrated. "With your line hand, you add in a fast crisp pull as you begin your back cast and then pull once again on the forward cast. The goal is acceleration and then a sudden stop. While the motion of your rod hand serves to flex the rod,
pulling down on the line with your other hand puts pressure on the rod tip and bends it even more."
The Biomechanics of Fly Casting by Al Kyte, Ed.Dr., Univ. of California and Gary Moran Ph.D. University of San Francisco
"The
most effective haulers pulled the line back for greater distance than the other casters primarily
during the final, accelerated stages of loading. Thus they stopped the haul and released the line father back as well (figure 10)."
Double Hauling by Bruce Richards
"When a short, sharp haul is made in the middle of this long stroke, tailing loops are the result. It only makes sense that
a poorly-timed, sharp haul will cause the rod tip to collapse into a concave pathway, thus making the line 'tail'."