A good question, Sunspace! Look at any chemical plant you pass on the road, and you’ll probably notice all kinds of interesting techniques used in plant design. The spiral tubing outside a stack IS to accomplish enhanced heat transfer! The most common everyday example are fins added to electronic devices to assist in heat transfer. Fins come in a variety of types, and must be sized correctly to gain the desired effect.
With smoke stacks, the need to better heat transfer is for several reasons. For example, hotter fumes require a greater stack diameter since the gas is expanded. Cooled stack fumes will help save on materials. Also, if the fumes can be cooled rapidly, a stack can be made of cheaper metals than if the stack fume temps are excessive. So, it can be very cost-effective in stack design to use this approach. Lower fume temps also means less thermal expansion of the materials which also reduces many headaches related to that aspect of stack design, too.
I digress, but as for spiral staircases, they are more commonly found around storage tanks. Storage tanks don’t just sit there…as one might think. They actually have equipment on the roof to help them to “breathe” (to vent). A storage tank “breathes” due to daily changes in temps, etc. Also, due to pumps moving fluids in and out of a storage vessel. Depends on if it liquid or gas storage. (Even liquid storage has some vapor above the liquid level which has certain venting, or breathing, requirements.)
Stacks don’t usually need access by foot, so stairways not always found around them. Some stacks may be found with a ladder or stairway leading to a catwalk platform encircling the stack - maybe located only midway up the stack. This is usually when a stack is directly off a specific vessel with its own burner… as opposed to the main stack(s) venting off the bi-products of the main process(es). In this case, the catwalk is granting access to the upper sections (zones) of the vessel - usually to access instrumentation such as thermocouples and pressure gauges.
By knowing process equipment design and piping, you can tell what processes any plant is using (in a general sense) when passing these plants along the highways, etc. The equipment arrangement and associated piping is kind of like the plant’s “signature”.
Plant design…neat stuff! 
“They’re coming to take me away ha-ha, ho-ho, hee-hee, to the funny farm where life is beautiful all the time… :)” - Napoleon IV