Blankets were an indispensable item for the men of the 2nd Regiment. They were regimental property, like muskets, and considered just as essential. They were also chronically in short supply. Proper heavy blankets were not widely made in South Carolina, and were thus usually imported from Europe. As the war ground on, supplies from overseas dwindled, and inflation skyrocketed. By 1780, blankets were the single most expensive item issued to the soldiers. For these reasons, each soldier was expected to personally carry his blanket when on the march, whether on campaign or simply transferring posts.
Native Americans are well documented to have used leather or web straps to carry large loads. These went by many names: tumpline, tompline, trumpline, topline, burden straps, hoppis, hoppees, hoppess, etc.; all of which mean roughly the same thing: a strap for carrying things. These could be slung over their shoulder, around the chest or over the forehead. When carrying blankets, the blankets were well-documented to have rolled the blanket around the strap, tying it up with the long ends of the strap or with separate straps, forming a bedroll.
During the French & Indian War, British troops frequently abandoned their clunky knapsacks in favor of the tumpline, especially Light Infantry. At one point, General Lord Howe actually had his entire army use tumplines: "Their haversacks were rolled up in a blanket, which they carried as did the Indians and Canadians." 1
At this point, I refer you to an excellent blog dealing with F&I and Revwar research entitled "Of Sorts for Provincials". The author has a well-researched post on the tumpline: http://ofsortsforprovincials.blogspot.com/2011/04/1-hoppis-david-hastens-tumpline.html
Note that about halfway through the post, there is a quote from the Journals of Henry Laurens; this documents that uniformed South Carolina Provincial troops used "tomplines" in 1763, at the close of the French & Indian War, only 12 years before the Revolution.
By the Revolution, the British were familiar with the use of straps to carry their blankets. As documented in the "Of Sorts" blog post, there are numerous references to British troops using what are called "blanket slings." These were made of hemp or linen webbing, and were sewn into some form of carriage to better or more efficiently secure the blanket than the tumplines used in the earlier war. In effect, the blanket sling was a military, European-ized version of the tumpline.
Other quotes in the "Of Sorts" blog post attribute tumplines to Continentals in Virginia and other states. Further documentation exists, including paintings: Xavier Della Gatta's painting of the Battle of Paoli  clearly shows Continental troops wearing their blankets rolled up on their backs; these are worn quite horizontally, suggesting a strap across the chest, rather than one worn over the shoulder (note Continentals casualties in the foreground). And Della Gatta's painting of the Battle of Germantown  shows the entire light infantry of the British 40th Foot wearing white blankets on their back in similar fashion to the Continentals at Paoli.
January 6 1778
"…The whole Reg.t are to be Powdered clean Shaved & in a Soldier like Dress in Order to Make a proper appearance in their march Throught the Town, their Blankets are to be neatly Roaled & fastned at their Backs, the Reg.t will land at Ropers wharf…" 7
March 1 1779
Camp at Purysburg
"…We rise here a little before Day break, the Men turning out with their Haversacks and Blankets on the Backs…" 8
These two records suggest that the men had their blankets rolled and strapped on their backs, not horse collar-style over their shoulder. This means that the men were provided with some sort of carrying strap. There is nothing in the records showing that the men were issued this sort of item. All expensive items, including leather shoulder carriages for bayonets and cartridge pouches, were carefully recorded, and soldiers were held accountable for their loss. Many necessary items, however, were provided to the men and not recorded at all: plates, cups, musket tools, etc. This suggests that, whatever the design of blanket straps, they were relatively inexpensive and fairly disposable. This eliminates the possibility of leather tumplines in favor of linen or hemp webbing and rope. Both materials were relatively plentiful in South Carolina throughout the war; the only notable clothing shortage shortages always revolved around the scarcity of wool, especially broadcloth.
With all this in mind, here is the arrangement we are adopting for blanket slings in the re-created 2nd Regiment:
1. The Journal of Captain John Knox: An Historical Journal of the Campaigns in N. America For Years 1757, 1758, 1759, and 1760. 3 Vols. (Toronto: The Champlain Society, 1914), vol. 1, p. 185.
2. della Gatta, Xavier. The Battle of Paoli. 1782. Reproduction from ExplorePAHistory.com. http://explorepahistory.com/displayimage.php?imgId=1-2-28F. Accessed February 21, 2012.
3. _____________. The Battle of Germantown. 1782. Reproduction from 4oth Regiment of Foot (American Revolutionary War reenactment organization). http://www.najecki.com/40thfoot/Purpose.html. Accessed February 21, 2012.
4. Atkinson, John. Private, 3rd Foot Guards, c. 1800 (Rear View). Watercolor on canvas. Prints, Drawings and Watercolors from the Anne S.K. Brown Military Collection, Brown University. http://library.brown.edu/cds/catalog/catalog.php?verb=render&id=1176496768796875. Accessed February 21, 2012.
5. de Loutherbourg, Phillip James. Embarkation to Corunna, 1808. Reproduction from Second Battalion, 95th Rifles Forum (Napoleonic War reenactment organization). http://2nd95thrifles.myfastforum.org/index.php?component=content&topicid=514&postdays=0&postorder=asc&start=30. Post 44. Accessed February 21, 2012.
6. Private photo. Second Battalion, 95th Rifles (Napoleonic War reenactment organization). http://2nd95thrifles.myfastforum.org/index.php?component=content&topicid=514&postdays=0&postorder=asc&start=75. Posts 83-84, 89-90. Accessed February 21, 2012.
7. Salley, A.S., ed. “An Order Book of the First Regiment, South Carolina Line, Continental Establishment.” The South Carolina Historical and Genealogical Magazine, 7-8 (1906-1907), p. 136.
8. Cross, Jack L. “Letters of Thomas Pinckney, 1775-1780.” South Carolina Historical Magazine, 58 (1957), p. 229.