I've had a look at Admiral Steel site.
The links seem to malfunction. Anyway, you can see that both cold and hot rolled steels are sold in various states.
You need the ANNEALED steels if you want them soft, as both cold and hot rolling can partially harden steel, apparently (I believe work hardening in one case, and hardening due to an abrupt drop in temperature in the case of hot rolling, but I'm just supposing that, I have no hard fact.)
When mild steel (iron) is concerned, HOT rolled is SOFT, COLD rolled is HARD (work hardening)