We have been down this road before as the above links suggest, ha, I think the second link actually appropriated our research, nothing wrong with that.
The modern stampede string seen at your local Corral West is not seen in the 19th century, the term is most likely 20th century nomenclature as I have never seen it in period writings. Excluding military use, tie downs for hats were seen but not that frequently amongst horsemen. Remember the use of a horse seldom requires breathtaking speeds with flying hooves and hats, few horsemen today have any need for a string on their hat. Watch a rodeo from roping to bull riding, those guy's don't use strings and more than half the time their hat remains on their head after the ride.
I did some work on this for my own information, I was not really that interested in border and Mexican culture, but here is what I found.
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The use of a chin strap to ‘tie down’ a military cap or shako on the wearer, and use of worsted wool hat cords for decoration and arguably to tighten an 1858 hat’s fit, is documented well before our time period. The existence of such items is in no way questioned. Rather, research attempts to place appropriate limitations on civilian use of a hat string or cord attachment for the purpose of securing a hat.
The necessity of this statement is subsequent to the popular use of what is known as “stampede strings” or “cords” by those involved with cowboy action shooting. Commonly available in western wear stores and providers of cowboy action shooting clothing, they generally are made of braded cord or heavy string, hang to the belt line, and often have decorative attachments to include an end that is made of horse tail hair much like a fly tassel used on horse tack. They are obvious when seen. Review of the literature and period images to date have not revealed any similar item from the later half of the 19th century. Further, the term “stampede string” or “cord” as yet has not been discovered in the writings of the 19th century.
A summary of what is known is given below.
Taken from I see by your outfit, by Lindmier and Mount: "Some Cowboy's used a leather thong which passed around the base of the crown and under the chin to keep the wind from blowing away their hats" John R Huhges, an early 1880's cowboy commented that the use of two "buckskin strings" to secure the cowboy's broad brim hat was common. However, the authors then go on to say, maybe in Montana but they could find no images of Wyoming cowboys using them. Note that this refers to two buckskin strings, and not a braided, decorative stampede cord.
Further from: The Unconventional History of Civilization on the Old-Time Cattle Range, by Philip Ashton Rollins (1869-1950), published 1936 "Around the crown (of the hat), just above the brim and for the purpose of regulating the fit of the hat, ran a belt which was adjustable as to length" this would assist in securing the hat but it is not a stampede string.
The author continues - "from either side of the brim at its inner edge, depended a buckskin thong; these two thongs, sometimes known as 'BONNET STRINGS' (my emphasis) being tied together and so forming a guard, which during rapid riding or in windy weather was pushed under the base of the skull, but which at other times was thrust inside the hat" This sounds familiar with two buckskin strings, but it sounds as if the author is describing it use on the (rear) base of the skull like a drill sergeants’ hat.
Further information is found in a book called: The Trail Drivers of Texas originally published in 1924 by George W. Saunders, President of The Old Time Trail Drivers' Association. The memoir named, "My First Five-Dollar Bill", contributed by J.L. McCaleb, takes place on a trail drive in 1868. McCaleb says "I wore a black plush hat which had a row of small stars around the rim, with buckskin strings to tie and hold on my head". It is quoted in Cowboy Culture, (top of page 220).
From Trail Drivers of Texas, by J. L. McCaleb, Carrizo Springs, Texas, p.486: "We bedded our cattle for the last time near Abilene, Kansas. The boss let myself and another boy go to the city one day. As it had been a long time since we had seen a house or a woman, they were good to look at. I wore a black plush hat which had a row of small stars around the rim, with buck-skin strings to tie and hold on my head. We went into town, tied our ponies, and the first place we visited was a saloon and dance hall."
These are the only sources to date that we have of "strings" attaching the hat to the head.
After looking at many, many images from the period we have none, NONE which shows an Anglo wearing anything similar to a modern stampede cord. Even amongst Californios or Vaqueros none are seen from the period wearing anything like the modern stampede cord.
What is occasionally seen is a simple functional string often of thin leather, but sometimes of ribbon tied with little more than 1 to 3 inches of excess material hanging beyond the tied knot.
From Cowboys and Trappings of the Old West, by Manns and Flood, page 27, there is an image showing a sombrero purportedly dating from the 1880's with a buckskin thong coursing the front of the crown and then piercing the brim at either side above the ear which is then tied with about three inches of excess thong below the tie. From the same source on page 113 there is an image showing a charro that is dated purportedly 1885 - it looks as if he has both a stiff military like leather strap and a string tied beneath his chin the string hanging to mid-chest. Again from the same source on pages 142 and 143 the cowboy riding third from left has a leather string coming from his small crowned short brimmed hat.
In the The Peacemakers Arms and Adventure in the American West, by R.L. Wilson, page 161 there is an image of King Fisher (killed in a gunfight in 1884) where what appears to be a leather thong attached to his hat and tied beneath his chin with about an inch of material hanging beneath his chin.
The image of Joe Cheeseman, attributed to the late 19th century and from the above source also, it shows a wide brimmed western hat – look closely and you will see a string placed behind the head.
There are other images, but notably when looking for such images one is struck by how seldom one sees any sort of hat tie. It appears to be limited to plainsmen or cowboys and infrequent if not uncommon in its use.
Relative to how the hat string attaches to the hat: there is no period reference or image of a civilian hat with a braided hat string where the braiding separates as it goes through the hat creating a circle of braided string on either side, both intended to envelope the crown of the hat - with exotic horse hair bangles hanging from it as the modern stampede strings show. Rather, most period strings seem to have gone through the hat preceding either in front of the crown or behind the crown, then through the brim on the opposite side and most often tied beneath the chin (no sliding apparatus) with perhaps an inch or two of extra sting hanging below.
Therefore lacking further evidence the following statements can be made:
1/ There is no such period nomenclature as stampede cord, stampede string, or stampede anything, but rather a thong, or bonnet string, or hat string.
2/ There is no evidence to suggest that a braided fancy cord hanging to the belt buckle with lots of decorative stuff existed in the 19th century.
3/ There will, on occasion, be seen a buckskin or ribbon string that would most likely be stuck inside the hat or cast behind it when not in use (excluding military chin straps).
4/ When considering either period photo's and/or written description, the wearing of such a bonnet string may have well been regional, certainly rare, and under no circumstance similar to that which is worn today and termed a stampede string. Their use by Anglo plainsmen from Texas to the Canadian border, cowboys of the Oregon basin, and Anglos of the southwest is infrequent if one considers all period photos of horsemen - there must be hundreds of photos failing to show bonnet strings for everyone where one can be found. If common they should be frequently seen, and they are not. They must be considered infrequent if not uncommon. Point in fact, their uncommon use continues to this day amongst working cowboys although seen with some frequency amongst modern California and Great Basin horseman who claim Buckaroo influence.