Similar to the improved double slit method you dig out alternate sides of a chamber and use the aquifer to drain itself. By offsetting the pumps you avoid backwash, save 1 tile and avoid rebuilding stairs.
The technique uses lower aquifer levels to drain the upper levels, so at some point all middle levels become self-draining.
This method requires a pump, wall building material and a miner who is at least a novice swimmer, and a pump operator. My miner is usually close to legendary by mid summer from mining out a dirt hovel. I used 2 pumps, but you can replace the second pump with a wall to isolate the output of the pump from the input, vertically mirror the second chamber and then move the pump when the instructions say to pump from the other side.
The method is broken into two parts: a technique for the final 2 layers and a mid layer technique for all of the rest. Always dig the exploratory UD stairs first so that you know which of the following sets of instructions to follow.
Here we go.
Top Level
Urist McMiner is an outstanding miner and novice swimmer so he will never drown regardless of the water depth or duration of immersion. His masonry is enabled, but only dabbling. All other dwarves are in burrows excluding the watery areas, otherwise the most foolish dwarf will drown admiring the spray from under 7 ticks of water, then the most altruistic dwarf will die attempting to retrieve the body and 2 more and 4 more ...
1. Dig 2 updown (UD) stairs below each of the down (D) stairs in the picture.
2. Channel out the input side of the pumps.
3. Build your pumps or your pump plus wall.
4. Depending on whether 2 levels down is aquifer or not follow the mid level instructions or the last two level instructions.
Mid Level
The mid level instructions are a cycle that you keep repeating until you reach the second last level of aquifer.
1. Start pumping from the east.
2. Dig out UD stair and the UD stair below it at (a) in the figure.
3. Dig out the 4 channels and the dig square near (b) in the figure.
4. Swap the pump.
5. Dig out the UD stairs, 4 channels and dig tile near (c).
6. Deconstruct the pump(s) and dig out the 2 D stairs and 2 dig tiles near (d). Everything should look like the second panel in the figure and be self draining.
7. Build 2 walls at (f) on the floor above, or if it is the top floor do nothing.
8. Build 2 walls at (e). These are the only 2 walls that will suspend and need to be unsuspended until the job is done.
9. Build the 6 walls and 2 pumps in the third panel and then everything should look like the fourth panel. Now you can bring in the non-swimmers to man the pumps.
Last Two Levels
In the figure below the top two panels are the second last level, and the bottom panel is the last level, incomplete and completed. It assumes that the exploratory stairwell that determined this is the last two levels is on the east(right) side.
Second Last Level
1. Start pumping from the east.
2. Dig the 4 channels, the UD stair, the UD stair below it and the dig tile near the (a). Also dig the one channel beneath the (c) in the figure.
3. Swap the pump.
4. Dig the 3 channels, 3 D stairs and UD stair, but not the UD stair on the level below and the one dig tile on the (b) side.
5. Dismantle the pump(s) and dig out the 1 D stair and 2 dig tiles in the center wall.
6. Build the 2 walls over the pump inputs on the level above.
7. Build the 3 walls over the D stairs under (b) and (c) in the figure. This walls will suspend and need to be resumed.
8. Build the remaining walls and pump(s) until it looks like the top right panel in the figure.
Last Level
1. Start and suspend an UD stair at (h) so that Urist can complete the UD stair after the 4 walls are completed.
2. Build the 3 walls at (e), (f) and (g). These will suspend.
3. Dig out (j) (east wall near (h)) and build a wall. This will suspend.
4. Unsuspend and complete the UD stair. Build a D stair above it so Urist can be free of the pit.
5. Dig 2 levels down and you are through.