I'd go with an extra, extra fine ground. I think espresso is a fine ground
My understanding is that the difference you want is in the *roast*, not the *grind*. The "rule of thumb" I have heard is that the longer the water is in contact with the grinds, the *coarser* the grinds.
An espresso or Turkish grind is intended for a situation where water is forced through the grounds quickly (an espresso machine, etc.). For a toddy cold brew you want *coarse* grinds.
If you were extracting with hot water, I agree.
But the problem the OP had with the cold water is that not enough of the coffee flavour is being extracted. Therefore a finer ground will allow more flavour to be removed.
The extraction will be more efficient with a higher temperature, higher pressure, finer grounds and longer time.
Espresso has temp, pressure and fine grounds. Drip coffee has temp and time.
Of course a perfectly efficient extraction is not what you actually want, as you don't want all the bitterness etc.. So you want a balance of the temp, pressure, fineness and time to extract the right flavours.
Now I'm sure this is all googleable for coffee, but going off general chemistry principles, if it's too weak, finer grounds will help.