This is usually due to the amount of dirt and perspiration the sweatband is designed to absorb.
Just to post my standard practice (and it's by far NOT the ONLY way, but it's worked for me all these years) that may help in that area.
Take a cotton ball and soak it in rubbing alcohol. Not dripping, but wet. Proceed to wipe the sweat band all the way round. Don't worry about the back of the band, just the side that touches your head. You'll be absolutely amazed how dirty your cotton ball will be. Do this with a couple of alcohol soaked cotton balls, but don't keep doing it until you have a completely clean cotton ball. If you go TOO deep with energetic cleaning, you may be pulling out the dye from the leather!

Alcohol will dry out leather, so once it's dry, you need to apply some leather treatment. A lot of folks advise using Pecards, and though that's a great product due to it's pH balanced formula, keep in mind, standard Pecards is a water resistant product. We still want our sweat bands to do their job, and this is to absorb, so we really don't want a repellent applied to the leather.
I use Pecards leather lotion (also advertised as 'motorcycle leather lotion') that is water soluable and is a cleaner/conditioner. A close product to this is Lexol leather cleaner/conditioner. This replaces the moisture lost from the alcohol cleaning, and also cleans and conditions the leather (yeah, well 'duh'!


After a couple light applications of the leather lotion (I just apply with my fingers rather than a cloth), allow it to completely air dry. Once dry, take a clean dry cotton rag and buff the band to remove any excess product and bring the band to a shine.
This adds years to the life of your sweatband, and also removes the grime from day to day wear as well as the odor of a 'well used hat'.
Just a tip offered.

Regards! Michaelson