I wear a size 7 3/4 hat. Meaning a very large head. Will this fit?

My cowboy hat size is 7 3/8 this hat rides up on top my head. So no! one size does not fit all.

I wear a 6 7/8 will this hat fit me ?

I’m a 6-7/8 also. The inner band and straw are flexible, so it does fit a little snug at first but it is forgiving. I wear mine around the lake and pool so a little snug is fine for me. Not that getting it wet is a good thing for the hat, but with wet hair it loosens up fine. It shrinks back as it dries. This is my 3rd hat of the same style/brand and I like it. I got 2 summers out of each one without having to be overall gentle with it.

Why is there a grommet on the brim in the back side of hat?e

You can put your chin strap up over the brim and then slide the strap through that hole until it reaches the wooden cinch bead . And it keeps the chin strap from dangling around, or coming off the brim. Some people like to wear the chin strap over the sides of the brim, this grommet makes sure it stays there. Pretty good idea if you ask me. They probably placed it to the side. So you don't have a hole in the center of the rear brim. This is just a guess