Do I need a jacket if I am only outside for ten minutes?

Often not, in dry, still conditions above roughly 50°F, because you will barely have cooled down before you are back inside. In wind or rain, ten minutes is long enough to matter.

What should I wear waiting for a bus in the cold?

Dress for standing still, which is warmer than dressing for walking the same distance. Gloves and something over the ears make more difference than a heavier coat.

Why do I feel cold at first and warm later on a walk?

Walking generates heat continuously, and it takes roughly ten minutes for that to offset the initial loss. Dressing for the first minute is the usual reason people end up overheated.

What to Wear for a Short Walk Outside

For a walk of about twenty minutes, dress for the conditions rather than for the temperature you would need all day: one layer lighter than you would choose for a long stint outdoors, because walking generates noticeable heat within roughly ten minutes. Below about 45°F, cover the extremities first — hands, ears and neck — since those lose heat fastest and cost the least to protect.

Duration changes the answer more than people expect

Almost every piece of clothing advice assumes an unstated duration, and it is usually a long one. But the same conditions ask for different clothing at five minutes, twenty minutes and three hours — and the difference is not small.

Time outside	Around 50°F	Around 35°F
5 minutes	Whatever you have on	Coat, nothing else needed
20 minutes	One light layer	Coat plus gloves
1 hour	Light jacket, closed shoes	Coat, gloves, hat
3 hours or more	Jacket, plan for the temperature dropping	Coat, gloves, hat, covered ears, warm socks

Two things drive this. Walking produces heat continuously, so a body that is cold in the first minute is often comfortable by the tenth. And discomfort in the extremities accumulates with time rather than arriving at once, which is why gloves matter at an hour and barely register at five minutes.

Cover the extremities before adding bulk

Below roughly 45°F, hands, ears and the back of the neck are where cold is felt first. Adding gloves and a hat changes comfort more than adding a heavier coat over a torso that was already warm enough — and it costs far less to carry.

This is also the part people most often get wrong in the other direction: a heavy coat with bare hands is a common and uncomfortable combination on a short cold walk.

When the general rule fails

You are waiting rather than walking. Standing at a stop generates no heat. Dress for standing still, not for the walk either side of it.
It is raining. Twenty minutes is long enough to soak through. Shedding water matters more than insulation at any temperature above freezing.
There is real wind. Wind removes the heat walking produces about as fast as you make it, which cancels most of the duration allowance.
You are going somewhere you cannot adjust. If the walk ends in a long stay outdoors, dress for the stay.

Conditions can become genuinely hazardous rather than merely uncomfortable. This page is about comfort. For cold or heat that carries a health risk, follow your national weather service — in the United States, the National Weather Service cold safety guidance.