Also, when you have a little money, make things like hats, bumper stickers, keychains, etc. People like to show off that they support things, especially if those things involve children. Even if they don't wear the hat, even if they don't use the keychain, people will still buy them. We're just stupid like that.
Another thing with the 'adoption' is that you wouldn't want people being left out. I know hospitals do this, but I'm not sure how. Depending on the size of the hospital, you could really lower the morale of the kids who don't get 'adopted'.
Field trips for the kids who can go out? Maybe take groups of them out to get ice cream? You could have them all wear matching tee shirts (if you have that kind of money) with the name/hospital. Ice cream and walking advertisement. Win win situation.
If you're really gutsy, get some taggers to graffiti the logo on some local buildings. Against the law, sure, but still advertisement.
Magazine sales, perhaps? Fundraisers are annoying for everyone, but people tend to order more magazines than they can read.