Does milk need to be cold before steaming?

Yes, straight from the fridge, and the reason is timing rather than temperature itself.

The aeration window

Milk stops accepting air at somewhere around 37 °C. After that, no amount of stretching adds foam — you are only heating.

Starting cold at 4 °C gives you a long run up to that point. Starting at room temperature gives you a few seconds, which is not enough to position the wand, judge the sound and stretch deliberately.

What that means in practice

Milk from the fridge, used immediately.

A cold jug helps for the same reason. Some people keep their pitcher in the fridge; rinsing it in cold water is enough.

Do not pre-warm the milk, ever. It is a common instinct and it removes your working time entirely.

Leftover milk

Milk that has been out on the worktop for twenty minutes will foam noticeably worse than the same milk from the fridge. Pour what you need and put the bottle back.

Milk that has been steamed

Cannot be re-steamed usefully, cold or not. The proteins have already done their work.

The related mistake

Positioning the wand and then turning the steam on wastes the window. Get the tip just below the surface first, then open the valve — the first seconds are the ones that matter.