When looking for a Tilley one should take a look at their various offerings to see which style and brim size they like best. I have a Tilley LTM6 (T6?) that is good for full sun but found it a little wide for stuffing into a backpack. So I now mostly use a T5 as the brim is narrower but more than enough for the brightest part of the day. I believe the T3 And T5 have similar brim widths, with the T3 slightly wider on the sides.I highly recommend a Tilley T3. It also helps to slather ears, neck, and sides around ears. I like Coppertone Sport.
A couple of years ago I went through every page on their website to try and make sense of their hat numbering and letter scheme (T1 vs T3 vs LTM5 ...) and brim sizes. I wrote it down on a scrap piece of paper but wished I had published what I found here, as their website is not very customer friendly in that aspect and I lost my secret decoder ring. I think they rely on a person buying one of their popular styles at some retail outlet or ordering some hat they have heard about.