Thanks all. I thought any cycling vest would stop the wind (it seems to when I drape it over a fan). And I thought that would be sufficient. Live and learn.

I will try a non-cycling fleece vest that I happen to have to see if it works before I consider spending more money.

Note to Electrik: I almost always do wait at least 30 minutes before I shower. It lets me upload my Garmin data, fill out my log, and decompress a little. But your point about warming up too quickly makes sense.