Get some Ballistol.
You can use it staight or diluted to clean the barrel and then swab the bore really good with it straight to prevent rust.
If you do the bucket thing, swab the bore good with a well-soaked straight Ballistol patch or bore swab. Even if you failed to get all the water out, Ballistol will mix witn the water and still protect the bore.
You can even put it on your boo-boos, although in the U.S. it's not approved for that.
