I also have a Tilley, and wore it more when sailing. I like a lightweight hat, and have a Columbia, which has light mesh around the crown, with single layer on the top, and a brim broader than the tilley--great for the summer. For wet weather, I have an OR in grey--which is Gortex lined on the brim and top/crown--the sides will velcro up, and it is good rain protection as well as sun. All have "strings".
Most of the time in the boat, it is a single billed cap. Up north-this is replaced by a billed cap with fleece ear flaps (at night, even in the summer--softie from Florida gets cold ears--again with Gortex on this cap-)