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."