I'm not familiar with SPI, so I'll take a look into it and see how it works.
I'd really like it to be at least 16 pixels in height.
That's the exact display board I'm trying to create, in monochrome of course
I've tried looking up the HT1632C driver, but I don't think it's a standard breadboard size IC.
I would recommend looking on instructables for LED Matrix display. There are numerous amounts of instructables on that very topic, at least last I checked.
Even if you wire them as 8 rows of 48 you can place them as 16 rows of 24.
Is it as simple as plugging in a higher power source into the arduino?
No.The good news is that each of the 48 columns will only have one LED turned on at a time. The row outputs will need to drive up to 48 LEDs at a time so that's nearly 1 Amp (0.96) if each draws 20 mA. A resistor per column will be needed to limit current.