Partially daisy chain shift registers
There is not an easy way to chain the modules at non-byte boundaries with that kind of shift register.

If you can tolerate using a different shift register, such as the 74HC164, shown below, then you can use any parallel output as the input to the next stage, thus skipping the bits beyond the selected output.  Note that if you are shifting at say 1 MHz, you can clock thousands of serial bits before the flicker becomes noticeable.



If you have an application where you want to serially load a pattern and then present it all at once by clocking the output register, then you could just add two unused bits at the start of the string that you are serially loading for each daisy chained module.  These would get shifted into the unused bits at the end of the module.