I just put on a short sleeved white t shirt, like a vest really, and a couple of old long sleeved tops, reasonably thin. Went out for a walk and you know what? They're fine, I was warm. I don't think I'll need to bother with a compression top. Not for now anyway, I'll see how it goes. I'm just going to buy a nice looking zip up or pullover blue or black hoodie as my outer garment.
(Even if it's just 10 or 15 minutes, it will give you a good idea of if you have the right gear or not
)
It's just that I want the rest of my gear first. But yeah, I'll start tomorrow. 
which seems a bit unfair.