As your E6400 has the lower rating PSU and it's 13 years old, it could be the beginning of a power supply failure ( the old bird is fine as long as you don' t over tax her)
Over time the capacitors weaken. You might find telltale signs of this by removing the outer casing and the PSU casing. What you are looking for are capacitors that are bulging or rounded on top.
If so you might find that the drive requirements are just too much for the PSU.
You could try using a laptop drive like the WD beve IDE ones, as they only pull about 2 watts instead of the 5 watts of your current desktop model.
Or instead You could try a card reader, but I don't know the PSU drain of these (although I would imagine it to be very low as there are no moving parts), however this will cost perhaps in excess of £100 with no guarantee of success, whereas you could pick up a WD Beve ATA HD from eBay for about £30.
Regarding the Frankenstein approach, the extra PSU could be fitted internally (bags of room inside) feed from the existing power cord, and not be at all obvious. This solution would extend the life of the old power supply by reducing the burden placed on it.
Or you could just source a new PSU with a bit more welly and all should be good for expansions for many years.
So many choices ...