A man went out to buy cigarettes at night in one spring day and he thought
春だから上一枚で煙草買いに出たけど夜は結構冷えるなぁ
I am not sure what does 上一枚で means exactly. I think 枚 counts number of cigarette boxes and 上 is a prefix attached to counter to indicate upper limit...?
枚 counts flat items. Boxes of cigarettes, would not be counted using 枚. However, it sounds to me like he went out with only one layer of warm clothing because he thought, "it's spring". But, it turned out to be colder than he expected. Without any further context, my guess is that 上 here refers to something to wear as a top layer: a sweater or a light jacket.
A sweater is something that might be counted using 枚 since they form a layer when worn.
上一枚 is definitely an outer garment, as A.Ellett says. He didn't think he would need go in multiple layers as it was already spring, but it was unexpectedly cold in the evening. He probably put something on over a t-shirt or whatever he would have been wearing inside, but it was not enough to feel comfortably warm.
As to the 上 part, in English we think of putting on more clothes as "over" or "on" each other, but when you dissect those words carefully, you'll realize that they also carry a latent figurative element of the physical "above," exactly the same as 上 in this case. 上 is "outermost."
