More digits, same three wires

The ninth bit out

Eight clock pulses fill the shift register. A ninth pushes the first bit off the end — and rather than losing it, the chip presents it on a pin called QH′.

Wire QH′ to the next board's SER and nothing is lost at all. The two registers become one sixteen-cell chain with a seam in the middle, and the same SRCLK and RCLK drive both.

Send the far digit first

This is the part that catches people. Every clock pulse pushes everything one place further along, so after sixteen pulses the byte you sent first has travelled furthest — through the near board, out of QH′, and into the far one.

For a clock showing 12:34, that means sending the 1 first and the 4 last. Reverse it and you get 43:21, every wire correct, and an hour spent checking jumpers.

digitalWrite(PIN_RCLK, LOW);
shiftOut(PIN_SER, PIN_SRCLK, MSBFIRST, segOut(kDigits[tens]));   // far board
shiftOut(PIN_SER, PIN_SRCLK, MSBFIRST, segOut(kDigits[units]));  // near board
digitalWrite(PIN_RCLK, HIGH);   // both digits change together

One RCLK pulse, after both bytes. That is what makes the pair update at the same instant instead of a fraction of a millisecond apart.

What each extra board costs

Three wires from the Arduino, still. SER goes to the first board only; SRCLK and RCLK go to all of them.

Eight more clock pulses per update. An Uno's shiftOut takes roughly a hundred microseconds a byte, so six digits is well under a millisecond. You would have to be updating thousands of times a second before this mattered.

A digit's worth of current from your 5 V supply. Each board has its own 74HC595 carrying its own digit, so the per-chip limit does not get worse as you chain. Your supply is what has to grow.

The one number that changes is QH′'s own drive: the datasheet specifies it for 4 mA rather than the 6 mA of the segment outputs. It is feeding the next board's SER pin, which draws almost nothing, so it never comes up in practice — but it is why QH′ is not a ninth segment output.

Twelve boards in the box

Twelve driver boards and twenty-four digits means you can build a six-digit clock and still have half the parts spare. The chain does not care how long it is; the thing to plan is the supply and the enclosure, not the wiring.