I have just made a bucket hat for my cheeky one year old. There is no way he would keep a hat on if it didn't have straps on it. I have worked out a way to keep the hat reversible but also have straps. I used buttons and straps with button holes. I have described it in detail here with some pictures to help - http://www.sewtogether.com.au/1/post/2011/03/oliver-s-reversible-bucket-hat-with-a-little-addition-attaching-straps-to-a-reversible-hat.html
I hope you find it useful!