The word "stampede string" is a modern term but I don't know what they were called back in our period. In Texas they probably had some kind of exotic Mexican sounding name for them like, La Stringerro de Sombrerro or something.
I have used three different ways to attach a stampede string to my hats.
Method 1: Punch holes through the brim next to the crown and above the ears. Run the thong around the outside of the crown in the front or the back and down through the holes. This is probably the most common way you see it done. It is the method that the Army drill instructer hats use. This method has some drawbacks. If you use your stampede string a lot, the hole will waller out or tear. I had a hatter put some small grommets in the holes and they worked for a while and then tore out. Also if your wear your hat in the rain or snow you will get an annoying drip, drip, drip down on your ears akin to the "Chinese Water Torture."
Method 2: Run the thong around on the inside of the hat between the sweatband and the crown and down through the seam above the ears. It works if you use a flat thong or a small round thong. Not good for the fatter briaded thongs. The Pros: No annlying drip or torn felt. Cons: It does not lend itself to fancy thongs. Also a big Oklahoma sized gust of wind can tear the stiching out leaving you with a sweatband around your head while you watch the body of your hat skip across the prairie. I remedied this by casting a couple of reinforcing stiches in the area where the thong passes between the stiching.
Method 3: Use a cotter pin pushed up between the sweatband and the crown and attach a thong to each side using a ring. I found this style at River Junction. They have a braided leather one and a horse hair one. I use the braided leather one. It looks good, is not scratchy like horsehair and no holes in the brim. I have not found any drawbacks yet. It has stood up to some pretty heavy winds without any apparent damage to the hat. If you can braid leather it would be easy to make. The only thing I don't like about the River Junction model is that it is too long for my likes.
Back to Hat Treatments: Found in
The Cowboy Hat Book, Revised Edition by William Reynolds and Ritch Band
....for oily stains...try Fuller's Earth, found in Drug or paint stores. It has the consistency of baking powder. Sprinkle the Fuller's Earth on and around the spot to a depth of about one-eigth of an inch. Like a poultice draws infection from a sore on your horse, this little poultice of Fuller's Earth draws the oily residue from the fibers in your hat..... The book is a good source for old photos, hat styles and care and feeding of your hat. There is a picture on page 46 of Ritch Rand using a hat sizing contraption on the head of a customer. The hatter I went to in Oklahoma City used one on me. That is where I found out my head was not a perfect long oval but more like a whopperjawed stove pipe. The contrapation creates a template of your head that the hatter uses to cut a hat form to match your skull.