When you buy your under-seat bag for your tools and tubes, it's well worth it to get a fairly large one. You might be able to put your jacket (depending on the jacket) into the under-seat bag.
If the jacket is too big to fit in a big under-seat bag, you can roll the jacket up with the sleeves sticking out and tie it around your waist. (keeps the jacket from flapping behind your and whapping into things)
I have a big bag on each of my bikes (roadie has a big handlebar bag, the commuter has a pannier) because I bring lots of stuff with me. On the roadie I will also usually wear a little fanny-pack. The fanny-pack has come in handy for strapping things to my back, like a new pair of handlebars I bought and shuttled home and a bike saddle I was taking to someone's house.
"If Americans want to live the American Dream, they should go to Denmark." - Richard Wilkinson