Between 2013 and 2017, I visited a dozen museums and saw more than 100 caps. Here is a list of the museums**, and the artifacts I saw at each one.

Each is listed with its museum dates. The eighteen deemed 18th C for this project are starred, and linked to their posts in this blog. I also wrote about one 19th C cap. These posts are linked out from this page.
Many more leads could not be followed up because of time constraints. More caps with eighteenth-century dates exist than were supposed, and, one assumes, more will be discovered.
#49.370 (infant)
#51.238 (infant)
#45.787 (infant)
#99.848 1820’s
Daughters of the American Revolution Museum, Washington DC
#78.65.9 1805
#82.138.9 1820
#2005.13 n.d.
#2464.2 1800
#1817 n.d.
#98.54.5 n.d.
#4510.2 n.d.
Kent State University Museum, Kent, Ohio
#1995.017.0467 1860
#1995.017.0399 1860
#1983.001.1493 1840
#1995.017.0455 1860
#1990.086.0002 1840
New Canaan Historical Society, CT
#CE 237 n.d.
#E169 19th C
#HS 19th C
#269B 18th C (Dutch)
#CE 173 18th C
#CE 164 18th C
Smithsonian National Museum of American History, Washington, DC
(Which is not the same thing as the Cooper Hewitt, Smithsonian Design Museum, in NYC. Their textiles section was under construction at the time of this study, so I didn’t get to see their artifacts.)
#274535.2 late 18th C
#220965.20 late 18th– early 19th C
#6608 1780-1800*
#1971.1162.28 1800-1899
#254326.2 late 18th –19th C
#372.213 n.d.
#312686 n.d.
#6609-B 1790-1810
Charleston Museum, Charleston, SC
#HT1836 1820-1830
#HT1693 1820-30
#HT2908 (1722-1837)
#2008.029.002 n.d.
#HT 4196 (a, b, & c.) n.d.
#2008.029.003 n.d.
#2006.041.003 Early 19th C
#HT6815 n.d.
#HT6392 1820
#HT 1641 n.d.
#HT1650 n.d.
#HT1644 n.d.
#HT1867 Late 18th C
#HT1868 n.d.
#HT5884 n.d.
#HT 1839 1790
# HT1876 n.d.
#HT 1872 n.d.
#HT1694 1850
#HT 1710 1850
#HT6309 1857
#HT1881 early 19th C
Genesee Country Village and Museum (Susan Green Costume Collection in the John L Wehle Gallery)
#88.195 1825
#2001.026 1800-1850
#2004.102 1800-1850
#97.024 1815
Philadelphia History Museum at the Atwater Kent
Note: closed until further notice!
#87.35.138 1800-1860
#1000.179 n.d.
#1000.176 n.d.
#87.35.994 1820
Chester County Historical Society, West Chester, PA
#1989.1891 n.d.
#1989.1889 n.d.
#1989.1888 n.d.
#1989.1892 1800-1840
#1989.1890 n.d.
#1989.1912 1800-1840
#1989.1903 n.d.
#1989.1882 1800-1840
#1989.1883 1800-1840
#1989.1886 1800-1840
#1989.1897 1800-1840
#1989.1902 1800-1840
#1989.1962 n.d.
#1989.1893 1800-1840
#1989.1894 1800-1840
#1989.1896 1800-1840
#1989.1955 1800-1840
#1989.1914 1800-1840
#1989.1954 1800-1840
#1989.1953 1800-1840
#1989.1959 n.d.
#1994.3270 1790-1810* (in Fitting and Proper)
XX #1989.1995 1790-1810* (in Fitting and Proper – did NOT see this cap) XX
(one item without a number, squared headpiece, 18th C?)
#1989.28.11 n.d.
1990.21.3 n.d.
Winterthur Museum, Garden, and Gallery, Delaware
Henry Whitfiled State Museum, Guilford, CT
#942020 18th C or 1827
#955003D n.d.
Musee McCord Museum, Montreal, Quebec
M987.32.1 1785
M987.32.2 1785
M965.136.86 early 19th C
M980.4.26 late 17th-early 18th*
**The Costume Institute at the Metropolitan Museum of Art, NYC, the Cooper Hewitt, Smithsonian Design Museum, NYC., and Historic New England, (fromerly Society for the Preservation of New England Antiquities (SPNEA)), Haverhill, MASS have caps with eighteenth-century dates, but these 3 museums were undergoing renovations during the period of this study.
Final.