@RedRover . I found this 1824 article of a deserter wearing the old 1811 "Glazed Cap". So the corps most likely started using the "flat tops" between 1824 and 1827.
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Again, super find.
Keep in mind that a "hat" is not a "cap", though many in recent times confound the two. There's a reason for employing one term or another. A "hat" had a brim. A cap did not, though it might have something like a partial brim in the form of a visor.
So lets see what a "glazed hat" meant at that time:
From 1820, Britain:
Charleston Daily Gazette and Advertiser, 11-13-1815:
From the Charleston, SC, Courier, 11-30-1822:
From 1801:
Regarding an American sailor in the Battle of Lake Champlain, 1814:
"Mr. James Sloane, of Oswego, informed me that, a few days before the battle, he gave one of the seamen a v
ery nice glazed hat. After the battle was over the sailor came to him with the hat in his hand, having a semicircular cut in the side and crown made by a cannon-shot while it was on his head. "Look here, Mr. Sloane," said the sailor, "how the damned John Bulls have spoiled my hat." He did not seem to reflect for a moment how nearly the cannon-ball came to spoiling his head."
A British seaman, described in the late 1820s:
N.H. Hazeltine, Letters from Europe, 1829.
British Navy midshipmen evidently wore "glazed hats" in lieu of their uniform hats in foul weather...
Southern Patriot, Charleston, SC, 7-21-1819:
American Beacon, Norfolk, VA, Nov. 10, 1820: "Glazed hat" with a brim:
Looking through the documents from the time, we can see that "glazed hats" by the 1820s were not uncommonly made of leather, that they were mass produced for servants, or other outdoor wear, and not infrequently worn by seamen too.
in 1811, the US Marine Corps adopted a leather "fatigue cap" of a folding type, to replace common hats then worn for fatigue wear. After the War of 1812-15, there is this document about a new "hat" proposed to replace the fatigue leather cap:
October 11, 1818:
The box, directed to you, contains a Hat; which the Inventor, the person, who furnished our new Uniform Caps, is desirous of substituting in the place of our common Leather Fatigue Caps. Hats of the model now sent, can be furnished at a very little more cost, than the Leather Caps, at present in use; they will certainly be much more serviceable and ornamental, for the Sea, as well as show service. Will you endeavor to have them adopted for the Corps? "
Brevet Major Richard Smith at New York to Brevet Major Archibald Henderson.
I have never seen reference to evidence the Marine Corps actually procured these new hats for fatigue (made by the same vendor who made the Corps' leather uniform caps, so probably of leather themselves). The notice you found, from 1824, gives a Marine in his gray fatigue dress, and with a "glazed hat" in lieu of a cap. So perhaps an evidence that the Marine Corps did buy hats for fatigue wear after 1818 of the type suggested above.
In any case by 1825 the Corps was once again to employ a "fatigue cap" and not a hat:
May 10, 1825, order by Col. Henderson: That in future, the following amount of clothing (and no more) will be issued to each Marine, for, and during the period of his enlistment. Viz. 5 uniform Coats with wings, 10 Pairs Uniform Woolen overalls, 10 Pairs Uniform Linen overalls, 20 Shirts, 2 Stocks, 20 Pairs Shoes, 3 Blankets, 2 Knapsacks, 10 Pairs Socks, 4 Pairs Gaiters, 5 Fatigue Jackets, 5 Fatigue Overalls,
1 Fatigue Cap, 1 Uniform Cap, plate, & Scales, 2 Cockades & Buttons, 5 Pompons, 5 Tassels, Great Coats will be allowed as follows, annually. For the Guard of a 74, 6. For the Guard of a Frigate, 4 For the Guard of a Sloop, 3. For the Guard of a Schooner, 2. For the different Stations, according to the number of men mounting Guard.
My conclusion would be that you have found an evidence that between 1818 and 1825, (particularly 1824) the Marine Corps was providing "glazed hats" for fatigue wear in lieu of the leather fatigue cap adopted in 1811. From May, 1825, however, if not before, a "fatigue cap" was provided again. This last was certainly the "flat topped" one, given the 1827 image from the naval yard, and referenced thereafter...