I myself would use something bigger than a wool dauber. I myself would use a sponge, that has more coverage area, per pass.
I would also have my dye in a dish like a disposable pie pan. I would dip one side of the sponge in the dye real quick like & pull it back out & kinda let the dye soak into the sponge. That way it don't drip. Drips off of the sponge is a bad deal & can wreck a project.
Then i would start my dye application process to the project, & continue to repeat till I got my desired results. Uniform color & desired color depth.
Try dying some of your scrap leather you cut your hat out of first. That way you have a idea how the dye is going to react to the leather.